Orbital parameterizations of the relativistic two-body problem due to Brumberg, Damour-Deruelle, Epstein-Haugan, and Blandford-Teukolsky as well as osculating elements are compared. Exact relations between constants describing the orbit in the parameterizations are derived. It is shown that all the parameterizations in question are valid not only in general relativity, but in a generic class of relativistic theories of gravity. The obtained results provide us with an additional check of consistency of different models used in timing of binary pulsars.