This paper studies $t$-interleaving on two-dimensional tori. Interleaving has applications in distributed data storage and burst error correction, and is closely related to Lee metric codes. A $t$-interleaving of a graph is defined as a vertex coloring in which any connected subgraph of $t$ or fewer vertices has a distinct color at every vertex. We say that a torus can be perfectly $t$-interleaved if its $t$-interleaving number (the minimum number of colors needed for a $t$-interleaving) meets the sphere-packing lower bound, $\lceil t^2/2 \rceil$. We show that a torus is perfectly $t$-interleavable if and only if its dimensions are both multiples of $\frac{t^2+1}{2}$ (if $t$ is odd) or $t$ (if $t$ is even). The next natural question is how much bigger the $t$-interleaving number is for those tori that are not perfectly $t$-interleavable, and the most important contribution of this paper is to find an optimal interleaving for all sufficiently large tori, proving that when a torus is large enough in both dimensions, its $t$-interleaving number is at most just one more than the sphere-packing lower bound. We also obtain bounds on $t$-interleaving numbers for the cases where one or both dimensions are not large, thus completing a general characterization of $t$-interleaving numbers for two-dimensional tori. Each of our upper bounds is accompanied by an efficient $t$-interleaving scheme that constructively achieves the bound.