Is there any automated way to test that code is compatible with both Python 2 and 3? I've seen plenty of documentation on how to write code that is compatible with both, but nothing on automatically checking. Basically a kind of linting for compatibility between versions rather than syntax/style.
I have thought of either running tests with both interpreters or running a tool like six or 2to3 and checking that nothing is output; unfortunately, the former requires that you have 100% coverage with your tests and I would assume the latter requires that you have valid Python 2 code and would only pick up issues in compatibility with Python 3.
Is there anything out there that will accomplish this task?
There is no "fool-proof" way of doing this other than running the code on both versions and finding inconsistencies.  With that said, CPython2.7 has a -3 flag which (according to the man page) says:

Warn about Python 3.x incompatibilities that 2to3 cannot trivially fix.

As for the case where you have valid python3 code and you want to backport it to python2.x -- You likely don't actually want to do this.  python3.x is the future.  This is likely to be a very painful problem in the general case.  A lot of the reason to start using python3.x is because then you gain access to all sorts of cool new features.  Trying to re-write code that is already relying on cool new features is frequently going to be very difficult.  Your much better off trying to upgrade python2.x packages to work on python3.x than doing things the other way around.
Checking code for compatibility with Python 2 and 3