We define and analyze the photometric orbit (PhO) of an extrasolar planet observed in reflected light. In our definition, the PhO is a Keplerian entity with six parameters: semimajor axis, eccentricity, mean anomaly at some particular time, argument of periastron, inclination angle, and effective radius, which is the square root of the geometric albedo times the planetary radius. Preliminarily, we assume a Lambertian phase function. We study in detail the case of short-period giant planets (SPGPs) and observational parameters relevant to the Kepler mission: 20 ppm photometry with normal errors, 6.5 hr cadence, and three-year duration. We define a relevant "planetary population of interest" in terms of probability distributions of the PhO parameters. We perform Monte Carlo experiments to estimate the ability to detect planets and to recover PhO parameters from light curves. We calibrate the completeness of a periodogram search technique, and find structure caused by degeneracy. We recover full orbital solutions from synthetic Kepler data sets and estimate the median errors in recovered PhO parameters. We treat in depth a case of a Jupiter body-double. For the stated assumptions, we find that Kepler should obtain orbital solutions for many of the 100-760 SPGP that Jenkins & Doyle estimate Kepler will discover. Because most or all of these discoveries will be followed up by ground-based radial velocity observations, the estimates of inclination angle from the PhO may enable the calculation of true companion masses: Kepler photometry may break the "msin i" degeneracy. PhO observations may be difficult. There is uncertainty about how low the albedos of SPGPs actually are, about their phase functions, and about a possible noise floor due to systematic errors from instrumental and stellar sources. Nevertheless, simple detection of SPGPs in reflected light should be robust in the regime of Kepler photometry, and estimates of all six orbital parameters may be feasible in at least a subset of cases.