Homotopy type of $X\times S^1/x_0\times S^1$

Question

The reduced suspension of X is defined as $$ \Sigma X=S^1\times X/S^1\vee X $$ I'm wondering what will happen if I only shrink one $S^1$ to a point.

In which condition can I have $S^1\times X/S^1\times x_0\simeq X\vee \Sigma X$?

Thanks.