  We analyze a theory known as extended DBI, which interpolates between DBI and
the $U(N)\times U(N)/U(N)$ non-linear sigma model and represents a nontrivial
example of theories with mixed power counting. We discuss symmetries of the
action and their geometrical origin; the special case of SU(2) extended DBI
theory is treated in great detail. The revealed symmetries lead to a new type
of graded soft theorem that allows us to prove on-shell constructibility of the
tree-level S-matrix. It turns out that the on-shell constructibility of the
full extended DBI remains valid, even if its DBI sub-theory is modified in such
a way to preserve its own on-shell constructibility. We thus propose a slight
generalization of the DBI sub-theory, which we call 2-scale DBI theory. Gluing
it back to the rest of the extended DBI theory gives a new set of on-shell
reconstructible theories -- the 2-scale extended DBI theory and its
descendants. The uniqueness of the parent theory is confirmed by the bottom-up
approach that uses on-shell amplitude methods exclusively.
