In this work, we generalize to hypercubic tilings of dimension n the description of tiling semigroups as inverse semigroups associated to factorial languages and the representation of this semigroup as a P*-semigroup. In addition, we show that, in contrast with the one-dimensional case, the tiling semigroup of any n-dimensional hypercubic tiling is always infinitely presented (even as a strongly E*-unitary inverse semigroup) and give a necessary and sufficient condition for two hypercubic tiling semigroups to be isomorphic.