Rolling a Die over a Labeled Board

Label the faces of a unit cube with numbers $1$--$6$ as in a die. Place the cube to sit on an integer lattice grid, with one corner at the origin and sides aligned with the axes. Completely label every lattice square of a rectangular \textquotedblleft{}board\textquotedblright{} $R$, whose corner is at the origin, with numbers in $\{1,2,3,4,5,6\}$. The problem is to roll the cube over its edges so that, for each square $s \in B$ labeled $l$, the cube lands on $s$ precisely once, and when it does so, the top face of the cube has label $l$.

What is the computational complexity of solving an instance of this problem?