  The sneutrino resonance reactions p \bar p -> sneutrino -> l^+ l^- +X, for
l=e,\mu,\tau, in the MSSM without R-parity are considered. We perform a
cross-section analysis and show that present limits on some products of
R-parity violating couplings in the sneutrino sector can be significantly
improved in future upgraded Fermilab Tevatron runs. Furthermore, we introduce
CP-violating and CP-conserving \tau-spin asymmetries which are generated
already at the TREE-LEVEL in the reaction p \bar p -> sneutrino -> \tau^+
\tau^- +X if there is muon-sneutrino - \bar muon-sneutrino mixing and that
vanish in the SM. We find, for example, that for muon-sneutrino masses in the
range 150 GeV < m_{muon-sneutrino} < 450 GeV, these spin asymmetries reach ~ 20
- 30% and ~ 10% for mass splitting between the muon-sneutrino CP-odd and
CP-even states at the level of \Delta m ~ \Gamma and \Delta m ~ \Gamma/4,
respectively, where \Gamma is the muon-sneutrino width. Both the CP-violating
and the CP-conserving spin asymmetries should be detectable in future Tevatron
runs even for a heavy sneutrino with a mass < 500 GeV. If detected, such
asymmetries - being proportional to the mass splitting between the CP-even and
CP-odd sneutrino states - may serve as a strong indication for the existence of
the sneutrino mixing phenomenon.
