I'm not sure Brexit really fits the previous right/left explanation does it? As you say, some parts of both camps voted against type, with Labour voters from smaller towns voting to leave, and Tory voters from bigger cities voting to remain. In that sense, if you're looking for clearer trends, it's cities/countryside or old(er)/young(er), maybe even graduates/non-graduates.
The problem as I see it is that Brexit has been a catch all for such a wide range of discontent and it's resulted in fudging. We see it clearly in Labour as Corbyn attempts to appeal to both metropolitan remainers and northern leavers by saying as little as possible, but it's undoubtedly more damaging in the Tories as they're running this whole thing. We're 600 days into this and there is still no semblance of what kind of Brexit they are really gunning for. This stems from the campaign itself, because it was such a dumbed down in/out vote, people couldn't voice their opinion on just what Brexit meant for them. Was is about migration? Was it about control? Was it about national pride? Was it about trade? Right now it could be all of those (or none of them), and so we've got various parts of the cabinet trumpeting different forms of Brexit and there's no cohesion at all. Even today Merkel said she's still curious as to what kind of Brexit we want.
As it is I wouldn't be at all surprised if the Irish border doesn't result in an 'out' in name only that is something akin to what the Swiss have, and whilst that may have benefits it's hard to imagine it really satisfying either side and simply kicks the can down the road.