Let $z=\cos x + i\sin x$. Then $$z^2 +\frac{1}{z^2}+z^4+\frac{1}{z^4}+z^6 + \frac{1}{z^6}=-2$$ Substituting $w=z^2 + \frac{1}{z^2}$ and simplifying yields: $$w^3+w^2-2w=0\implies w (w+2)(w-1)=0\implies w = 0, 1, -2$$ When $w=0 \implies z^2+\frac{1}{z^2}=0 \implies z^4 = 1 \implies x = 45^\circ, 135^\circ, 225^\circ, 315^\circ$. When $w=1 \implies z^2+\frac{1}{z^2}=1 \implies z^4 -z^2 + 1=0\implies z^2 = \cos 60^\circ + i\sin 60^\circ, \text{or} \cos 300^\circ + i\sin 300^\circ \implies x = 30^\circ, 150^\circ, 210^\circ, 330^\circ$. When $w=-2 \implies z^2 +\frac{1}{z^2}=-2 \implies (z^2-1)^2=0 \implies z =\pm 1\implies x=180^\circ$. Hence we conclude $$x=\boxed{30^\circ, 45^\circ, 135^\circ, 150^\circ, 180^\circ, 210^\circ, 225^\circ, 315^\circ, 330^\circ}$$