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 ≲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.