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Tesselation (or tiling) generally implies translational symmetry, because you can generally move one part of a tiling over another a specific distance away and get an exact match (ie the tesselation is periodic). A counterexample (possibly the only one) is Penrose tiling, which is non-periodic. There is certainly no need for a tesselating shape to have either bilateral or rotational symmetry: all triangles and all parallelograms (including squares and rectangles) will tessellate.

I'm afraid this is a rather superficial answer to this very interesting question; a deeper one will have to come from someone with a knowledge of group theory.

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Q: How is symmetry related to tessellation?
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