  We study the magnetotransport of high-mobility electrons in monolayer and
bilayer MoSe$_2$, which show Shubnikov-de Haas (SdH) oscillations and quantum
Hall states in high magnetic fields. An electron effective mass of 0.8$m_e$ is
extracted from the SdH oscillations' temperature dependence; $m_e$ is the bare
electron mass. At a fixed electron density the longitudinal resistance shows
minima at filling factors (FFs) that are either predominantly odd, or
predominantly even, with a parity that changes as the density is tuned. The SdH
oscillations are insensitive to an in-plane magnetic field, consistent with an
out-of-plane spin orientation of electrons at the $K$-point. We attribute the
FFs parity transitions to an interaction enhancement of the Zeeman energy as
the density is reduced, resulting in an increased Zeeman-to-cyclotron energy
ratio.
