Drag the dot on the square — or play Snake on it — and watch what happens on the surface.
Drag the dot. Push it off an edge and it reappears at the mirrored spot on the opposite edge — that is the gluing rule. The hollow dot is the antipodal partner point.
Drag to rotate, scroll to zoom. The colours match the square, so you can see which patch goes where. Anything hidden behind the surface really is behind it.
The square is $[-1,1]^2$ with $p \sim -p$ on the boundary: the polygon $abab$ presenting $\mathbb{RP}^2$.
It is carried to $\overline{\mathbb{D}}$ by the odd map $$(u,v) \;\longmapsto\; w \;=\; \frac{\max(|u|,|v|)}{\sqrt{u^2+v^2}}\,(u+iv),$$ and then to $\mathbb{R}^3$ by the Bryant–Kusner parametrization of Boy's surface: with $D = w^6 + \sqrt{5}\,w^3 - 1$, $$g_1 = -\tfrac{3}{2}\operatorname{Im}\!\left[\frac{w(1-w^4)}{D}\right], \quad g_2 = -\tfrac{3}{2}\operatorname{Re}\!\left[\frac{w(1+w^4)}{D}\right], \quad g_3 = \operatorname{Im}\!\left[\frac{1+w^6}{D}\right] - \tfrac{1}{2}, \quad P = \frac{(g_1,g_2,g_3)}{g_1^2+g_2^2+g_3^2}.$$ Since $P(w) = P(-1/\bar w)$, this descends to $\mathbb{RP}^2$.