= Solution
Take periodic orbital <wavefunctions> on the ring, and write $H_0=-(\hbar^2/2I)\partial_\theta^2$. Between kicks the evolution over one period is
$$
U_T=e^{-iH_0T/\hbar}
=\exp\!\left(\frac{i\hbar T}{2I}\partial_\theta^2\right).
$$
Integrating through an infinitesimal kick gives
$$
U_\eta=\exp\!\left[-\frac i\hbar(-\hbar\eta\cos\theta)\right]
=e^{i\eta\cos\theta}.
$$
The free-motion contribution during the vanishing kick duration disappears in the ideal impulsive limit. The sign of the <phase> follows from the negative kick potential.
Starting just after a kick, the <particle> evolves freely first and is kicked at the end of the interval. Operators act right to left, so \b[one step is $U_\eta U_T$], and
$$
\boxed{U(0^+\to NT^+)=(U_\eta U_T)^N.}
$$
The one-period product is the <Floquet operator> of this <quantum kicked rotor>. There is no additional kick at the initial endpoint $0^+$. Changing the stroboscopic convention to just before each kick would change the displayed order; it is not appropriate for these endpoints.
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