In his "Theory of retracts", Borsuk defines the concept of h-dominance (h is for homotopically), that is: given two (in this post i consider only Hausdorff spaces) topological spaces $X,Y$ we say that $X$ h-dominates $Y$ if there is an h-map of $X$ into $Y$ and in this case we write $Y\underset{r}{\leq} X$.
We say that a map $f\colon X\mapsto Y$ is a h-map provided that there exists a map $g\colon Y\mapsto X$ such that $f\circ g$ is homotopic to the identity of $Y$. If $X\underset{r}{\leq} Y$ and $Y\underset{r}{\leq} X$ we say that these two spaces are h-equal.
My question is: does there exist two h-equal but not h-equivalent topological spaces? More explicitly i want an example (if there is one) of two spaces $X,Y$ not h-equivalent and maps $f_1,f_2\colon X\mapsto Y$ and $g_1,g_2\colon Y\mapsto X$ such that $f_1\circ g_1$ is homotopic to the identity of $Y$, while $g_2\circ f_2$ is homotopic to the identity of $X$ (if i didn't understand bad).