You say
In my view, the term 'alt-right' is almost a euphemism to legitimize an extreme right-wing ideology which only a few years ago would have been entirely outside the bounds of reasonable political discourse.
The "alt-right" are not further right than the far right. On many subjects, they are further left. For example, Steve Bannon:
“Like [Andrew] Jackson’s populism, we’re going to build an entirely new political movement,” he told Wolff. “It’s everything related to jobs. The conservatives are going to go crazy. I’m the guy pushing a trillion-dollar infrastructure plan. With negative interest rates throughout the world, it’s the greatest opportunity to rebuild everything. Ship yards, iron works, get them all jacked up. We’re just going to throw it up against the wall and see if it sticks. It will be as exciting as the 1930s, greater than the Reagan revolution — conservatives, plus populists, in an economic nationalist movement.”
Note how he rejects conservatism and refers back admirably to the 1930s, a time that defined the Democrats as the big spending party.
You say
They oppose immigration. They are nationalistic. They are opposed to globalization.
Other than the nationalism, these are also characteristics of left-wing populists like Bernie Sanders. The clearest sign of anti-globalization feeling is opposition to trade deals, and Sanders' rhetoric on trade is not that different from Trump's.
Sanders has also expressed skepticism on immigration as suppressing wages. He voted against immigration reform in 2007. This was also true of moderate Democrats like Jon Tester and liberal Democrats like Sherrod Brown. And thirteen Republicans voted for immigration reform in 2007. If all the Democrats had voted for it, it would have passed. At that time, immigration was not a partisan, left/right issue. Both support and opposition were bipartisan (even the independents split, Sanders against and Joe Lieberman for).
Are these positions where if you go far enough right, you get back to the left? Or vice versa? Or are these positions where they are mixing and matching between left and right? Like the way Bannon supports infrastructure spending, a traditionally left position.
While it is convenient to think of politics in terms of left and rightleft and right, it's not accurate. Politics are often more complicated than that. Someone can be in favor of a strong national defense and same sex marriage. Or low taxes and abortion choice. Note that in the 1970s, Republicans like Arlen Specter were pro-choice while Democrats like Bill Clinton and Al Gore ran on pro-life platforms.
Thinking about everything in terms of left and right can cause confusion. For example, the Democrats in the United States have similar positions (on average) to the Tories in the United Kingdom. Yet the Tories are generally considered a right wing party and the Democrats are considered left wing.
Left and right are not fundamental characteristics of politics. They are based on an observation about seating arrangements in one particular political circumstance. While labels simplify thinking (left good, right bad, or vice versa), they do not ultimately help it. What they do is help demagoguery and increase partisanship.