I have recently realized that in one of my (published) papers I have used the "inverse" numeration for the $H\mathbb{Z}$-homology of the objects of the stable homotopy category (so, if we consider the singular homology of the spectrum of a space it will be put in non-positive degrees). My (silly) question is: can one say that this is the "cohomological convention" for singular homology (or something like this) and denote this homology by $H_{i}^{sing}$, or is this numeration absolutely unacceptable? Certainly, the usual way to switch the numeration is to consider $H^i=H_{-i}$; yet in topology one never calls singular homology a cohomology theory!

See Homology or cohomology? for a certain discussion related to "my" convention.

Upd. Since $H_{i}^{sing}(X)$ for $\pi_{-i}X (H\wedge\mathbb{Z})$ is too misleading; I will probably write $H_{i}^{EM,\mathbb{Z} }(X)$ instead. Could you suggest something better? My problem is that I would not like to change the numeration for all homological (=covariant cohomological) functors in my papers (and I want to cite my previous papers in the new one).