Function Spaces over Products with Ordinals

We are concerned with the question of when a homeomorphism between $C_p(X\times \tau )$ and $C_p(Y\times \tau )$ implies the existence of a homeomorphism between $C_p(X\times {\tau }^{\mathrm{\prime }})$ and $C_p(Y\times {\tau }^{\mathrm{\prime }})$, where $\tau$ and ${\tau }^{\mathrm{\prime }}$ are spaces of ordinals with some additional desired properties.