  Rotation and orbital eccentricity both strongly influence planetary climate.
Eccentricities can often be measured for exoplanets, but rotation rates are
currently difficult or impossible to constrain. Here we examine how the
combined effects of rotation and eccentricity on observed emission from
ocean-rich terrestrial planets can be used to infer their rotation rates in
circumstances where their eccentricities are known. We employ an Earth climate
model with no land and a slab ocean, and consider two eccentricities ($e=0.3$
and 0.6) and two rotation rates: a fast Earth-like period of 24 hours, and a
slower pseudo-synchronous period that generalizes spin synchronization for
eccentric orbits. We adopt bandpasses of the Mid-Infrared Instrument on the
James Webb Space Telescope as a template for future photometry. At $e=0.3$ the
rotation rates can be distinguished if the planet transits near periastron,
because slow rotation produces a strong day-night contrast and thus an emission
minimum during periastron. However, light curves behave similarly if the planet
is eclipsed near periastron, as well as for either viewing geometry at $e=0.6$.
Rotation rates can nevertheless be distinguished using ratios of emission in
different bands, one in the water vapor window with another in a region of
strong water absorption. These ratios vary over an orbit by $\lesssim\!0.1$ dex
for Earth-like rotation, but by 0.3--0.5 dex for pseudo-synchronous rotation
because of large day-night contrast in upper-tropospheric water. For planets
with condensible atmospheric constituents in eccentric orbits, rotation regimes
might thus be distinguished with infrared observations for a range of viewing
geometries.
