Right, no universal rules, just have to make up a game that's self-consistent enough to have some appeal.
Whereas fiddling with the 5th postulate (about parallel lines) has been a standard way to branch to non-Euclidean geometry, another route was suggested by Karl Menger, dimension theorist, in his The Theory of Relativity and Geometry (1949).
Per this essay, we might define points, lines and planes to be "lumps" (as in "of clay"), not distinguished by their "dimension number" (standard ideas of "linear independence" between height and breadth and width, per French res extensa philosophy (ala Descartes)) is what's being undermined (swapped out) in this picture (we keep an idea of energy though, say "energy has shape").
By this time, we have a pretty coherent geometry built up, including lots of input from Euler's topology, sometimes share it with kids under the marketing label of 'Claymation Station' (for obvious reasons).
This way of thinking about geometry doesn't really interfere with most Euclidean proofs, so is hardly that radical a departure, just a different way of thinking and talking (alternative nomenclature, e.g. MITE for "minimum tetrahedron"). At Saturday Academy, we think of this as a sub-branch of Gnu Math, i.e. teach it under that category, running FOSS on commodity hardware. (FOSS = free and open source software).
More on math-thinking-l (open archive), also you can read about my meeting with Dr. Livio on this topic at our recent get-together @ Linus Pauling House in Portland, Oregon.