The Sierpiński product of graphs generalises the vast and relevant class of Sierpiński-type graphs, and is also related to the classic lexicographic product of graphs. Our first main results are necessary and sufficient conditions for the higher connectivity of Sierpiński products. Among other applications, we characterise the polyhedral ($3$-connected and planar) Sierpiński products of polyhedra. Our other main result is the complete classification of the regular polyhedral Sierpiński products, and more generally of the regular, connected, planar Sierpiński products. To prove this classification, we introduce and study the intriguing class of planar graphs where each vertex may be assigned a colour in such a way that each vertex has neighbours of the same set of colours and in the same cyclic order around the vertex. We also completely classify the planar lexicographic products.