Deep (>10 dB) long-duration (>1 ms) scintillation fades, caused by propagation through refractive-index turbulence, are the principal impairment that must be overcome to realize Gbps-class laser communication over line-of-sight atmospheric paths in clear-weather conditions. Spatial diversity reception can ameliorate such fades, to a degree, but current systems typically rely on forward error-correction and interleaving to achieve reliable communication over the atmospheric channel. This paper, together with its companion [A. L. Puryear, J. H. Shapiro, and R. R. Parenti, “Reciprocity-enhanced optical communication through atmospheric turbulence—Part II: Communication architectures and performance,” to be submitted to J. Opt. Commun. Netw.], comprise a two-part study that introduces and analyzes an alternative approach, in which atmospheric reciprocity is exploited to eliminate the need for interleaving and minimize the amount of forward error-correction required. The present work (Part I) first describes the problem setting and then presents proofs for reciprocity principles—with and without phase compensation—that apply under rather general conditions. By specializing to the far-field regime, the optimum (power-transfer maximizing) phase compensation is identified. These results underlie the communication architectures and performance analysis that will be reported in the Part II paper.