 Solution

Apply ${\sigma }_x$ to every odd site and use the first pulling-through identity on every two-site block. Adjacent $V_a^{-1}{V}_a$ factors cancel, while periodicity and cyclicity of the trace cancel the final pair. The MPS is unchanged. The identical argument with $V_b$ applies to every even site. Therefore the state has the two commuting physical symmetrieswhich generate ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$.