A famous locally gauge invariant quantity is
$$ F_{\mu \nu} = \partial_\mu A_\nu - \partial_\nu A_\mu \, , $$ which is interpreted as the measurable electric and magnetic field strengths.
Now, another locally gauge invariant quantity is
$$ C_\mu = \partial_\mu \varphi(x) - A_\mu(x) \, , $$ where $ \varphi(x)$ is the phase of the wave function $\Psi(x) = A e^{i \varphi(x)}$, which, for example, describes a single electron.
What's the standard interpretation of this gauge invariant quantity $C_\mu$?
A local gauge transformation is
$$ \Psi(x) \to e^{i \epsilon(x)} \Psi(x) $$ $$ A_\mu(x) \to A_\mu(x) + \partial_\mu \epsilon(x) $$