In quantum mechanics, a (unitary) symmetry is a unitary operator $U$ which commutes with the hamiltonian $H$ of the system:
$$ [U, H] = 0 $$
For instance, in the uniform Ising model with periodic boundary conditions:
$$ H = -J\sum_i\sigma_i^z\sigma_{i+1}^z $$
there is a parity symmetry $P$ and the translation $T$ symmetry (periodic boundary conditions are assumed):
$$ P = \prod_i \sigma_i^x,\quad T = \prod_k T_{k,k+1} $$
where $T_{k,k+1}$ is the exchange operator of two neighbouring spins. Indeed, this model also has an extensive amount of unitary symmetries, since spin-$1/2$ operators are both hermitian and unitary and are conserved quantities of the system:
$$ [\sigma_i^z,H] = 0,\;\forall i $$
In general, starting from the spectrum of $H$, one can construct "infinite" unitary symmetries of the system, by linear combination of the projectors times a phase:
$$ H|\psi_{\alpha,\nu}\rangle = E_{\alpha,\nu}|\psi_{\alpha,\nu}\rangle \rightarrow \begin{cases} U = \sum_{\alpha,\nu} e^{i\phi_\nu} |\psi_{\alpha,\nu}\rangle\langle\psi_{\alpha,\nu} | \\ [U,H] = 0 \end{cases} $$
where the spectrum of $H$ has been divided into sets labelled by an index $\nu$.
Is there some sense in which it is meaningful to consider only certain symmetries and not others? The only one I can think of is that $U$ is a local unitary operator, i.e. it can be written as the evolution operator a local hermitian operator $V = \sum_i V_i$ (where $V_i$ acts only on some finite region of the system and not all $V_i$ are the identity):
$$ U = e^{-i\sum_i V_i} = \prod_i e^{-i V_i} $$
For instance, this is the case for the parity symmetry and translation symmetry of the Ising chain, but not for $\sigma_i^z$.
There is also the problem of "equivalent" symmetries. For instance, there is a sense in which the two following symmetries are both parity symmetries of the Ising model:
$$ P = \prod_i\sigma_i^x,\quad P' = \prod_i\sigma_i^y $$
since they both flip the spins along the $z$ direction.
I'm asking this question because, for instance, in many-body systems, one is often interested in "perturbations which preserve the symmetry". Of course, one is not interested in perturbations which conserve every symmetry of $H$ (by the definition above), but rather the "physically meaningful" symmetries. Is there some sense which allows one to identify such symmetries and "equivalent" formulations, especially for many-body quantum systems?
Furthermore, it is particularly important in Landau's symmetry breaking paradigm of phase transitions, which says that spontaneous symmetry breaking is always accompanied by a phase transition.
Therefore, another, possibly more precise statement of my question, although somewhat different, is: what are the symmetries whose "spontaneous breaking" lead to a phase transition?
