  The muonic hydrogen experiment measuring the $2P-2S$ transition energy [R.
Pohl {\em et al.}, Nature {\bf 466}, 213 (2010)] is significantly discrepant
with theoretical predictions based on quantum electrodynamics. A possible
approach to resolve this conundrum is to compare experimental values with
theoretical predictions in another system, muonic deuterium $\mu$D. The only
correction which might be questioned in $\mu$D is that due to the deuteron
polarizability. We investigate this effect in detail and observe cancellation
with the elastic contribution. The total value obtained for the deuteron
structure correction in the $2P-2S$ transition is 1.680(16) meV.
