In terms of status, I think the left and right would tend towards different prestige games, but that it's not the full picture.
To save people scrolling back, Storr's prestige games are "success" (involving symbols of achievement) and "virtue" (involving demonstrations of superior morality).
Right and left economy
On the face of it, right-wing economy places more obvious value on working hard and achieving success—i.e. the assumption of meritocracy. Under both meritocracy and capitalism, individuals are celebrated for playing the success game. If you get a promotion or an expensive car, it means you worked hard for it.
Left-wing economy is more to do with moving the collective towards success, and individual success games become less possible the further left you go. The virtue game has to take precedent, and in economic terms this is largely a display of political allegiance. The Soviets, the Corbynites and the Blairites played this game to differing degrees. Soviet citizens could gain status by grassing up their comrades for not being sufficiently party devoted; Corbyn supporters might freeze out members of their social group.
Conservatives and progressives
Neither of these political philosophies are inherently right or left, and I'd actually argue progressivism is hyper-individualistic, but they tend to be associated with current party values.
Conservatism kind of combines success and virtue, because your symbols of achievement are reflections of your good work ethic. In the US this position is associated more with Christianity, so it has another virtue angle there.
Progressivism does kind of combine both in the sense you might lose your success symbols if you aren't sufficiently virtuous (just look at all the ongoing employment tribunals) but I think it's much more about virtue.
There's also liberalism, but I don't feel knowledgeable enough to make a stab at where that might fit in.
Individual v collective
This is the part I haven't explored, which is whether you can play a collective success game. On the world stage, I suppose you can. So in that respect, a lot of left-wing economies would be success economies.
But as a general tendency, I think right-wing individuals would probably lean more success (while playing a less obvious virtue game at the same time) and left-wing individuals would tend to lean more virtue (with some shared glory in party success).
Obviously this is mostly hypothetical and as Perking has pointed out, may not translate so well to the "messy" real world.