  Double polarization elastic e-p cross sections and asymmetries are considered
in collider kinematics. Covariant expressions are derived for the general
situation involving crossed beams; these are checked against the well-known
results obtained when the proton is at rest. Results are given using modern
models for the proton electromagnetic form factors for kinematics of interest
in e-p colliders such as the EIC facility which is in its planning stage. In
context, parity-violating elastic e-p scattering is compared and contrasted
with these double-polarization (parity-conserving) results.
