What is the surface obtained by identifying antipodal points of $\mathbb{S}^1 \times \mathbb{S}^1$?

This is perhaps a soft question.

Let $X=\mathbb{S}^1 \times \mathbb{S}^1$. Let $\mathbb{Z}_2$ act on $X$ by setting $(-1) \cdot (\theta,\psi)=(\theta+\pi,\psi+\pi)$. Consider the quotient space $X/ \mathbb{Z}_2$ which is obtained after identifying $ (\theta,\psi) \sim(\theta+\pi,\psi+\pi)$.

Is there a succinct description of $X/ \mathbb{Z}_2$ as some product or twisted/fibered product or something like that?

Are there other "simple" descriptions of this space? Is it related to some projective space?

I feel like there should be a "right" terminology to describe it, or a way to recognize it as some familiar space, but I fail to see it.

I understand that identifying antipodal points on the $2$-torus embedded in $\mathbb{R}^3$ results in a Klein bottle- but this is not the same identification we are doing here:

Here we identify $(\theta,\psi)=(\theta+\pi,\psi+\pi)$, and in the embedded description we identify $(\theta,\psi)=(\theta+\pi,-\psi)$.

from Hot Weekly Questions - Mathematics Stack Exchange