The Hexic transform ρ of the noncommutative 2-torus Aθ is the canonical order 6 automorphism defined by ρ(U)=V , ρ(V)=e−πiθU−1V , where U , V are the canonical unitary generators obeying the unitary Heisenberg commutation relation VU=e2πiθUV . The Cubic transform is κ=ρ2 . These are canonical analogues of the noncommutative Fourier transform, and their associated fixed point C ⁎ -algebras Aθρ , Aθκ are noncommutative Z6 , Z3 toroidal orbifolds, respectively. For a large class of irrationals θ and rational approximations p/q of θ , a projection e of trace q2θ−pq is constructed in Aθ that is invariant under the Hexic transform. Further, this projection is shown to be a matrix projection in the sense that it is approximately central, the cut down algebra eAθe contains a Hexic invariant q×q matrix algebra M whose unit is e and such that the cut downs eUe , eVe are approximately inside M . It is also shown that these invariant matrix projections are covariant in that they arise from a continuous section E(t) of C∞ -projections of the continuous field {At}0
