@iDave - In European countries, where elections are proportionally representative, each government is formed out of a coalition several political parties post-election (in general, the biggest party gets to pick the prime minister, not a rule... but important positions get allocated according to size). The voters can identify these parties when they cast their vote and the parties can identify their voters as well, so there's less of a guessing game of "who actually voted for Party X" and "what politics the voters actually voted for". It also seems to me that people are flexible about switching parties. This is why it took me almost a year to understand that the easiest way to predict what an individual US voter will vote is to look at what they voted last time. Shirt-color collections are a robust method in a de facto two-party system. It's not in a 10+ party system.
One of the most interesting thing about the fracturing of the GOP and now to the Democrats is that each US 'coalition' (Conservatives, Tea Party, Libertarians, Trumpists, Social Democrats, Democratic Socialists, Greens) have visible mirror parties in each European country. Implying if the US system wasn't split into two giant party blocks, the veil would be lifted from the voting demographic. Because of the way proportional representation works, center parties who are willing to switch allegiance from left to right tend to hold a disproportional influence. (Compare to centrist US senators who will sometimes vote for and sometimes against their own party).
It also makes it possible to see when a party runs a program that manages to recruit substantial numbers of people from other parties and where these voters come from!
I'm mostly familiar with Danish national populists which have also grown substantially in power (the "Danish People's Party" are currently part of the governing coalition). They are usually associated with the far-right... but it's more accurate to see them as syncretic ideologies which if you want to be cynical about it is not really an ideology at all and which if you want to be open-minded about it tries to reconcile parts from many different ideologies. Or if you wanna be cynical again: basically they're just opportunists.
It all started with the "Progress Party" (Fremskridtspartiet) back in 1976. PP was initially based on libertarianism and anarcho-capitalism (their leader actually went to prison for tax evasion). The early leaders saw the US as an ideal example and their stated goal was to reduce the government's bureaucracy as much as possible (at the time, government expenditure had grown to record levels and government debt was beginning to enter runaway mode. This was later solved in 1986 with austerity programs) as well as reducing personal income tax to zero.
They won surprisingly and "bigly" (28/179 seats in parliament... no small player) in 1976. Because of the fast success, they didn't have much of any political organization (the party leader wasn't really into paperwork) and no real ideology anyway (just a bunch of ideas), and so the party floundered in direction. Lots of internal disagreement. I'm sure we can recognize much of this in the Trump Administration as well ("What do you mean I have to bring my own staff/administration to the White House when I become President?!").
Unlike the other parties with libertarian leanings (e.g. Venstre, which means left but is actually on the right being the neoliberals and the Conservatives), the PP were also eurosceptics (Dexiters?).
In the 1980s, PP added a strong nationalist component seeking to ban all further immigration from Muslim countries including deporting all current immigrants. This is what they became most known for in the general public and to some degree this and the dexiting would be the unifying thread going forward. But de facto they mostly served as an obstructionist/deregulation party whose main policy was being in opposition. Perhaps because of this + the lack of internal organization, the party fractured into those who believed that maybe being pragmatic and working with other political parties would be smarter and traditionalists (compare to the Freedom Caucus) who believed in holding the line against any new regulation.
This resulted in a giant clusterfuck in which the party splintered and spun off several smaller parties over the years. "The Free Democrats", "Freedom 2000", and "Danish People's Party" all of which were initially smaller than the parent-party. Only the latter survived. In 2006 there was another scandal when the chairman welcomed people from "Danish Front" (self-declared white supremacists) causing some members to quit in protest and the party splintered again. In any case, the party still exists but it has not held seats in parliament since 2001.
Now ... DPP is the more interesting one when they fractured out, they were down to 4/179 but they now hold 37/179 seats which makes them the second-largest party. However, they specifically DID NOT want to be in government perhaps wisely understanding that if they were they would be held accountable for the outcomes of their policies. Instead they occasionally support the government which is a coalition of Venstre (the neoliberals, see above) and the Conservative (compare those to traditional US conservatives minus the religion).
So what are the current policies, if any, of the DPP?
They're still primarily an anti-immigration (at some point they literally declared themselves as the anti-muslim party) and anti-euro party. This is also the dimension along which they manage to trade votes with the government getting various admissions to restrict immigration policies thus resulting in Europe's hardest immigration policies(*). If they could build a wall, they would. Basically the aim is to prevent a multi-ethnic country which for them means reducing non-Western immigration and mandating cultural assimilation (mandatory language tests and lessons, teaching Christian(+) religion in schools, ... ).
(+) Even as it has an official state religion, Denmark is one of the least religious countries in the world so this is mainly to piss off the Muslims.
In terms of social policies (and this is where the interesting syncretic part comes in) they aim to improve conditions for the elderly and the disabled. For this reason, they've won a over lot of voters from the left, in particular the "Social People's Party" (think of them as democratic socialists, sardonically known as the party of school teachers and social workers) and the "Social Democrats" which used to be the biggest party. (Maybe compare to what happened for Trump in Michigan et al. these would be the equivalent of Social Democrat voters going to the DPP.)
They also mandate zero-tolerance policies for criminals when it comes to things like rape, murder, animal cruelty, and DUI but I suppose for reasons of personal preservation, they want to reduce punishment for hate speech of which they tend to be quite tolerant. Go figure
(*) On a personal note, for a few years, it was literally not possible for DW and I to move to Denmark until earlier this year when new laws replaced the old ones after lots of pressure. There were several examples of spouses living in the country and getting deported or immigrants who had lived there for decades and getting deported because their "attachment" to the country was no longer considered strong enough under the tighter rules. All that !@#$ ended, but it certainly caused some damage to individual people and families for the several years it lasted. There was literally an example in the papers about a Danish astrophysicist who had married an American woman and sold their house in California moving the family, children included, to Denmark ... and then the rules changed and the wife gets a letter from immigration services telling her that the rules had changed and that the children could stay but she had to leave the country. That could easily have been us. This still could be us, but in the US.
In terms of voter statistics ... and this will sound very familiar to the US.
DPP increasingly attracts unskilled workers (which used to go to the Social Democrats) in proportion to the socioeconomic Gini index divide ("as the rich get richer" more and more hate the elites because they aren't elite themselves). The bigger the divide, the stronger the attraction. Traditional European/Danish politics primarily happens along the socioeconomic divide but with increasing refugee streams due to ME wars and climate change, there's also a split forming along the sociocultural divide which is becoming increasingly important and for some people more so than socioeconomics. Here DPP attracts people who culturally conservative and wants to keep things the way they were (back in the 1950s). Make the country great again, meaning back the way it was.
To bring it all back to the original question: ...
The traditional left-right continuum, where left=deregulate the culture and regulate the economy and right=regulate the culture and deregulate the economy required a 2D-diagram to differentiate the regulation but that doesn't really fit the nationalist/nativist populists as they currently are with Trump, DPP in Denmark, and FN in France, and so on ...
These are in favor of regulating the culture and regulating the economy, that is, in favor of regulation. We only need 1 dimension for that.
Historically, for most of the history of these parties, it has been about regulating the culture (but along nationalist/nativist lines rather than moral lines which is the traditional conservative approach). While they dabbled with free markets it is now realized that this does not benefit their voters, so now they favor economic regulation that benefits their cultural voters: protect the elderly and the unskilled workers (protectionism). What's crazy about is that while they are strong on regulation and in that sense quite anti-libertarian, they still speak the libertarian language of their roots and well enough ("No more taxes!") to get the traditional neoliberal conservatives to go along on economic issues, so...
PS: This would be a lot easier to explain with graphs