Noether's 2nd variational theorem gives a correspondence between symmetries of a Lagrangian and Noether identities = relations among the EL equations.
How about relations among relations among the EL equations cf. syzygies?
Noether's 2nd variational theorem gives a correspondence between symmetries of a Lagrangian and Noether identities = relations among the EL equations. How about relations among relations among the EL equations cf. syzygies? 


Identities among Noether identities (lets call them higher stage Noether identities, for lack of a better standard name) are mapped to gauge symmetries of gauge symmetries (lets call them higher stage gauge symmetries, also sometimes known as reducibility identities). As far as I know, references to the notions of higher stage Noether identities and higher stage gauge symmetries and to this result are scattered through, and sometimes hidden in, the literature on the BRST treatment of gauge theories. For instance, the construction of the KoszulTate differential in the BatalinVilkovisky (BV) treatment of gauge theories requires that each generation of ghosts (ghosts of ghost, ghosts of ghosts of ghosts, etc., which generate higher stage gauge symmetries) has corresponding antifields (which generate higher gauge Noether identities). I'm sure this is well known to you, Jim, since this well known review article by Marc Henneaux (Secs. 2.10, 5.4) mentions it and refers to one of your old articles for details:
A more recent reference that mentions the identification between higher stage gauge symmetries and higher stage Noether identities is Sec. 3.2 of
There is also a generalization of Noether's first theorem which puts higher stage Noether identities into correspondence with higher degree conserved currents (where the latter refers to onshell closed differential forms of lower formdegree). I can add references for that as well, if it is part of your question. 

