I've been thinking about this.
I have, in my time, worked in the civil service. Up to a certain grade, you're allowed to go on, say, political marches in your spare time (so long as you don't wear a t-shirt saying "DfT employees for [political party]"), write letters to the newspapers, lobby MPs, even stand for election as a local councillor (though not as an MP - you'd have to resign regardless of grade).
Above a certain grade (7 in old money I think) you can't do any of this, even in your own time, because your job is to implement government policy to the best of your ability (whether you agree with it or not) while "speaking truth unto power."
Personally I think the BBC, being publicly funded and set up according to its charter to be politically neutral, should apply similar standards to the civil service code to the journalists reporting on the news, newscasters, current affairs presenters and editors.
This policy shouldn't (and I don't think does) apply to back-room staff like camera people, sound engineers, researchers - anyone who's not on camera as the "face of the impartial BBC." But I do think it's fair enough to have rules for those on camera - it genuinely is part of their job description to be seen to be politically neutral. And that means no tweets, even in a personal capacity, or snapshots of them at [insert protest march of your choice].