[hamish_todd]

The geometric product does transform composition. It has all the properties you want for multiplication in an algebra, or indeed a monoid or group.

So it is like matrix multiplication, but for transforms represented as multivectors. Multivectors are nicer than matrices because they are made out of the separate (exterior algebra) objects so you can geometrically interpret them. For example, a rotation-reflection (rotoreflection/improper rotation) will have a grade 1 part and a grade 3 part. One of them is the plane you reflect in, one is the point you rotate around.