What is the sum of all possible values of $t$ between $0$ and $360$ such that the triangle with vertices $(\cos20^\circ,\sin20^\circ), (\cos100^\circ,\sin100^\circ)$, and $(\cos t^\circ,\sin t^\circ)$ is isosceles?

1. Isosceles configurations occur when $t$ is equidistant from both given angles ($t=60$ or $t=240$), or when $t$ matches the chord-length to one of them ($t=2(20)-100=-60\equiv300$, or $t=2(100)-20=180$).
2. The four valid values are $60,240,300,180$.
3. Sum $=60+240+300+180=\boxed{780}$.