Counting hypercube Tetris pieces
Python 2, 338 335 330

Try it online!

from itertools import*
n,k=input()
O=[([0]*k,)]
R=range(k)
exec"O={min(tuple(sorted(zip(*[[x-min(p)for x in p]for p in f])))for q in permutations(zip(*(o+(r,))))for f in product(*[[p,[-x for x in p]]for p in q]))for o in O for s in o for r in[tuple(s[i]+x*(i==j)for i in R)for j in R for x in-1,1]if not r in o};"*~-n
print len(O)


Reads n,k as a list from stdin (e.g. [4,2]) and prints the number of polyominoes.

Starts with a single polyomino with a single cell at the origin.

n-1 times, replaces the polyominoes with all possible polyominoes created by adding an adjacent cell to an existing polyomino, and canonizing that polyomino by finding the minimum of all possible rotations of it.

-6 bytes thanks to math junkie, -2 bytes thanks to Surculose Sputum.