  Using the original advanced version of the direct method, we efficiently
compute the equivalence groupoids and equivalence groups of two peculiar
classes of Kolmogorov backward equations with power diffusivity and solve the
problems of their complete group classifications. The results on the
equivalence groups are double-checked with the algebraic method. Within these
classes, the remarkable Fokker-Planck and the fine Kolmogorov backward
equations are distinguished by their exceptional symmetry properties. We extend
the known results on these two equations to their counterparts with respect to
a distinguished discrete equivalence transformation. Additionally, we carry out
Lie reductions of the equations under consideration up to the point
equivalence, exhaustively study their hidden Lie symmetries and generate wider
families of their new exact solutions via acting by their recursion operators
on constructed Lie-invariant solutions. This analysis reveals eight powers of
the space variable with exponents -1, 0, 1, 2, 3, 4, 5 and 6 as values of the
diffusion coefficient that are prominent due to symmetry properties of the
corresponding equations.
