Orbital-phase-resolved infrared spectra of Cygnus X-3 in outburst and quiescence, including tomographic analysis, are presented, We confirm the phasing of broad He II and N V lines in quiescence, such that maximum blueshift corresponds to the X-ray minimum at Phi = 0.00 +/- 0.04. In outburst, double-peaked He I structures show a similar phasing with two significant differences: (1) although varying in relative strength, there is continuous line emission in blue and red peaks around the orbit; and (2) an absorption component, similar to 1/4 of an orbit out of phase with the emission features, is discerned. Doppler tomograms of the double-peaked profiles are consistent with a disk-wind geometry, rotating at velocities of 1000 km s(-1). Regrettably, the tomography algorithm will produce a similar ring structure from alternative line sources if contaminated by overlying P Cygni profiles. This is certainly the case in the strong 2.0587 mu m He I line, leading to an ambiguous solution for the nature of double-peaked emission. The absorption feature, detected 1/4 of an orbit out of phase with the emission features, is consistent with an origin in the He star wind and yields for the first time a plausible radial velocity curve for the system. We directly derive the mass function of the system, 0.027 M.. If we assume a neutron star accretor and adopt a high orbital inclination, i > 60 degrees, we obtain a mass range for the He star of 5 M. less than or similar to M-WR less than or similar to 11 M.. Alternatively, if the compact object is a black hole, we estimate M-BH less than or similar to 10 M.. We discuss the implications of these masses for the nature and size of the binary system.