  We introduce and classify nonequivalent commensurate stackings for bilayer
dice or $\mathcal{T}_3$ lattice. For each of the four stackings with vertical
alignment of sites in two layers, a tight-binding model and an effective model
describing the properties in the vicinity of the threefold band-crossing points
are derived. Focusing on these band-crossing points, we found that although the
energy spectrum remains always gapless, depending on the stacking, different
types of quasiparticle spectra arise. They include those with flat, tilted,
anisotropic semi-Dirac, and $C_3$-corrugated energy bands. We use the derived
tight-binding models to calculate the density of states and the spectral
function. The corresponding results reveal drastic redistribution of the
spectral weight due to the inter-layer coupling that is unique for each of the
stackings.
