  We show that the current search for Lorentz invariance violation (LIV) in the
summed energy spectra of electrons in $2\nu\beta\beta$ decay can be extended by
investigating the single electron spectra and the angular correlation between
the emitted electrons. We derive and calculate the LIV contributions to these
spectra associated with the anisotropic part of the countershaded operator and
controlled through the coefficient $\mathring{a}_{\text{of}}^{(3)}$ and discuss
possible signatures that may be probed in experiments. First, we show that some
distortion occurs in the single electron spectrum, maximal at small electron
energies. Then, we show that other LIV effects may be highlighted by analysing
the angular correlation spectra and the ratio between the Standard Model
Extension (SME) electron spectra and their Standard Model (SM) forms. We found
that these LIV signatures depend on the magnitude of
$\mathring{a}_{\text{of}}^{(3)}$, manifest differently for positive and
negative values of this coefficient, and become more pronounced as the electron
energy approaches the $Q$-value. Finally, we propose an alternative, new method
to constrain $\mathring{a}_{\text{of}}^{(3)}$ through the measurement of the
angular correlation coefficient. Using this method, and considering only
statistical uncertainties, we obtain bounds of $\mathring{a}_{\text{of}}^{(3)}$
at the level of present ones, obtained from summed energy spectra. We show that
future experiments can improve these limits significantly. Our study is
performed for $^{100}$Mo, but the results hold qualitatively for other nuclei
that undergo a double-beta decay. We hope our results will provide additional
motivation for the LIV analyses performed in DBD experiments.
