  A complete set of parity- and time-reversal conserving relativistic
nucleon-nucleon contact operators is identified up to the order $O(p^4)$ of the
expansion in soft momenta $p$. A basis is also provided for the corresponding
non-relativistic operators contributing in the general reference frame. We show
that the non-relativistic expansions of the relativistic operators involve
twenty-six independent combinations, two starting at $O(p^0)$, seven at order
$O(p^2)$ and seventeen at order $O(p^4)$. This gives supporting evidence to the
existence of two free low-energy constants which parametrize an interaction
depending on the total nucleon pair momentum ${\bf P}$, and were recently found
to be instrumental for the resolution of the long standing $A_y$ problem in
low-energy $p-d$ scattering. Furthermore, all remaining ${\bf P}$-dependent
interactions at the same order are uniquely determined as relativistic
corrections.
