Let a charge $q$ enter a constant magnetic field $B$ at 90 degrees, hence radiating EM energy-momentum and spiraling inwards to some point as it slows down. Time reversing this, including the current generating the magnetic field so it now becomes $-B$, the charge still radiates outgoing EM radiation and spiral inwards. All the events are reversed compared to the original case, so why isn't the EM radiation and spiraling reversed?
It is time reversible, you just have to remember to time reverse the field too. Absorption is the time reverse of emission. In the fully time reversed situation, the charge absorbs coincidentally perfectly arranged incoming EM radiation, which causes it to spiral outward.
In the standard derivation of EM radiation, you find that to satisfy the equations, an accelerating charge has to either emit outgoing radiation or absorb incoming radiation, or a combination of the two. To get the standard result, we neglect incoming radiation, but in your situation you can't.