Is Boris Johnson the right's Ken Livingstone? It came to mind as a result of his unexpectedly strong remarks about housing benefit. A lot of Tories disbelieve that Johnson is genuinely committed to the party. Ken spent large chunks of his career either at odds with the Labour Party leadership or outside the party. Johnson is now reprising Livingstone's role in protesting against Thatcher, while also reprising his role a second time around as an alternative version of a government he's fundamentally sympathetic to.
A lot of people remarked that Ken Livingstone, as mayor, was remarkably keen on facilitating the City's interests for someone whose staff included John Ross. Johnson is heavily reliant on the remaining ex-Livingstone officials to keep City Hall's basic functions going. Both of them put a lot of effort into maintaining a public image that is almost a caricature of their party - the whole tedious Shower Jobby act, vs. all the stuff about newts and public transport.
Of course, this overstates a bit. But I do think there's a significant truth here, and I suspect that future Mayors of London are going to have more in common with Ken and the Jobby than they will with the Prime Minister of the day. They will tend to be noisy and brash, given to ranting, and drawn back towards consensus within London by the administrative realities. There is famously no Democratic or Republican way to collect the garbage*. However, they will also tend to operate in permanent tension with the national government up river. This is an expression of the structural factors - you can't position yourself politically by replacing the Underground with a network of cable cars over the streets or abolishing school, so you've got to do so by picking fights with Westminster.
Given that, you're either going to be in the role of unofficial opposition leader, or else aligned with the government of the day's rebels, whoever they may be. Also, it seems that you'll probably end up being a couple of points to the left of your party either rhetorically or operationally. Despite all the yelling, Ken Livingstone was basically following the Blairite "let the bankers rip and then do some redistribution" plan, but with more aggression and nous. It's also true of the Jobby - for all the bullshit, he's not actually changed that much, which puts him some way left of the cuts consensus. Interestingly, this also seems to be true of Bertrand Delanoe and Klaus Wowereit, and perhaps also Michael Bloomberg.
* This argument may no longer seem as convincing as it once did, as there are probably Republicans who want to abolish rubbish collection...