Is it mathematics or physics?

I think that Mathematical Physics as a research area struggles with its identity at times. Is it mathematics or physics or neither?
I don't think Chemical Physics has a comparable identity crisis, with its value being appreciated by both communities.
Many? theoretical physicists (particularly great ones like Feynman and Anderson) will claim that most big breakthroughs in theory occur using old mathematics and with little regard to mathematical rigour. Sometimes a focus on mathematical formalism is an impediment rather than a aid to real progress. (See for example, this post about Anderson's views).
Whether string theory is actually theoretical physics or just beautiful mathematics is debatable (Anderson had a nice review of Peter Woit's book a while back).

However, there are times where Mathematical Physics does really produce some nice new mathematics. A recent case is the work by Stanislav Smirnov on conformal invariance and was honoured by award of a 2010 Fields Medal.