:::::Linking some terms for easy access: equivariant map, principal homogeneous space, group representation. 
 :::::The terms might compactly tell you what you are referring to, Sławomir, but they are making my head spin. I have recently become a little familiar with a geometry defined in terms of a group action on a set (Klein's Erlangen program), but am not sure that this even relates to what you are talking about. I am most familiar with tensors defined in terms of a vector space and its dual space, and this does can be done in the absence of a metric tensor. Without further structure, such a vector space (and hence the associated tensor algebra) exhibits GL(n) symmetry, which contains all the O(p,q) groups as subgroups, but there is no natural way to identify any one of these without an associated quadratic form. It seem to me that there is no way to tie any specific orthogonal group to the general (nonmetric) tensor concept, though it is possible that I've missed what you are really saying. —