  Hexagonal MnTe emerges as a critical component in designing magnetic quantum
heterostructures, calling for a detailed study. After finding a suitable
combination of exchange-correlation functional and corrections, our study
within {\em ab initio} density functional theory uncovers an insulating state
with a preferred antiferromagnetic order. We compute the exchange interaction
strengths to estimate the antiferromagnetic ordering temperature via Monte
Carlo calculations. Our calculations and symmetry analysis reveal a large spin
splitting in the system due to the antiferromagnetic order without considering
spin-orbit interaction, except in the $k_x$-$k_y$ plane. Critically examining
the band dispersion and spin textures obtained from our calculations and
comparing them with an insightful symmetry analysis and analytical model, we
confirm a combined Rashba-Dresselhaus interaction in the $k_x$-$k_y$ plane,
around the K point of the system. Finally, we find ferroelectricity in the
system for a higher energy magnetic configuration. Our results and insights
would help design heterostructures of MnTe for technological applications.
