In Ogg's paper "rational points on certain elliptic modular curves", the author says, using Ogg's notation for cusps, that for fiexed $d$, if $(y, N) = d$, then for any $x$ satisfying $(x, y, N) = 1$, the cusps of the form $$\left( \begin{array}{c} x \\ y \end{array} \right) $$ are conjugate, and hence for $d = 1, 2$, these are rational.

But I don't know the moduli interpretation for cusps in $X_1(N)$ for $N$ composite.

I know that the cusps $$\left( \begin{array}{c} 0 \\ y \end{array} \right) $$ are corresponding to the Neron $N$-gon with a $\Gamma_1(N)$-structure $(1, a) \in \mu_N \times \mathbb{Z}/N$, and the cusps $$\left( \begin{array}{c} x \\ 0 \end{array} \right) $$ are $1$-gon.

How about other cusps?