Flipped Perfect Square

Since $\overline{dcbaa}$ is a perfect square, $a=1,4,5,6,9$ . It is well known that $n^2 \equiv 0,1 \pmod{4}$ , hence $\overline{dcbaa} \equiv \overline{aa} \pmod{4}$ , leaving $a=4$ as the only possible value of $a$ . Therefore $\overline{aabcd}=\overline{44bcd}$ . Note that $209^2=43681 < 44000, 210^2=44100>44000,212^2=44944<45000,213^2=45369>45000$ , hence $\overline{44bcd}=210^2, 211^2, 212^2$ . Now $210^2=44100$ . This fails since $d \neq 0$ . Then $211^2 = 44521, 12544 = 112^2$ and $212^2=44944$ , so both $211^2,212^2$ work. Thus the sum of all possible values of $\overline{abcd}=211^2+212^2=\boxed{89465}$ .