We study the polarization properties of elastic electron scattering on H-like ions using the relativistic QED theory. We calculate the complete set of parameters which describe the scattering of a polarized electron on an initially unpolarized ion and analyze them for both nonresonant and resonant scattering focusing on the LL-shell resonances. The study is carried out for
B
4
+
,

Ca
19
+
,

Kr
35
+
, and
Xe
53
+
ions. We demonstrate that for the resonant electron scattering on ions with
Z
up to 50 both the relativistic spin-orbit and the exchange interaction are equally significant. The possible spin exchange between the incident and bound electrons can fundamentally change the polarization of the scattered electron. We show that the involvement of the autoionizing states leads to quantitative and qualitative changes in the polarization parameters, in particular, to a significant increase in the spin asymmetry.