  We present a detailed estimate of the transverse muon polarization (P_T) due
to electromagnetic final-state interactions in the decay $K^+ \to \mu^+ \nu
\gamma$. This CP-conserving effect represents the dominant contribution to P_T
within the Standard Model. As a result of an explicit calculation, we find that
the CP-conserving contribution to P_T is quite small, typically of
$\cO(10^{-4})$, essentially due to the suppression factor $\alpha/4\pi^2$. This
enforces the sensitivity of P_T in probing extensions of the Standard Model
with new sources of time-reversal violations. A brief discussion about possible
CP-violating contributions to P_T in the framework of supersymmetric models
with unbroken R-parity is also presented.
