From $a + b = c + d$ we get $(a + b)^2 = (c + d)^2$, hence $ab = cd$ and furthermore $a^2 - 2ab + b^2 = c^2 - 2cd + d^2$. As $(a - b)^2 = (c - d)^2$, we have $|a - b| = |c - d|$, which implies $a - b = c - d$ or $a - b = d - c$. Recall that $a + b = c + d$, so either $a = c, b = d$ or $a = d, b = c$.
The numbers must have one of the forms $\overline{aaaa}$, $\overline{abba}$ or $\overline{abab}$, with $a \neq 0$ and $a \neq b$. In all, there are $9 + 9 \cdot 9 + 9 \cdot 9 = 9 + 81 + 81 = 171$ numbers.