  We show existence and uniqueness for small data of regular time-periodic
solutions to the Navier-Stokes problem in the exterior of a rigid body,
$\mathscr B$, that moves by time-periodic translational motion of the same
period along a constant direction, $\bfe_1$, and spins with constant angular
velocity $\bfomega$ parallel to $\bfe_1$. We also study the spatial asymptotic
behavior of such solutions and show, in particular, that if $\mathscr B$ has a
net motion characterized by a non-zero average translational velocity
$\bar{\bfxi}$, then the solution exhibit a wake-like behavior in the direction
$-\bar{\bfxi}$ entirely analogous to that of a steady-state flow around a body
that moves with velocity $\bar{\bfxi}$ and angular velocity $\bfomega$.
