  Given a function f: {0,1}^n \to {0,1}, the f-isomorphism testing problem
requires a randomized algorithm to distinguish functions that are identical to
f up to relabeling of the input variables from functions that are far from
being so. An important open question in property testing is to determine for
which functions f we can test f-isomorphism with a constant number of queries.
Despite much recent attention to this question, essentially only two classes of
functions were known to be efficiently isomorphism testable: symmetric
functions and juntas.
  We unify and extend these results by showing that all partially symmetric
functions---functions invariant to the reordering of all but a constant number
of their variables---are efficiently isomorphism-testable. This class of
functions, first introduced by Shannon, includes symmetric functions, juntas,
and many other functions as well. We conjecture that these functions are
essentially the only functions efficiently isomorphism-testable.
  To prove our main result, we also show that partial symmetry is efficiently
testable. In turn, to prove this result we had to revisit the junta testing
problem. We provide a new proof of correctness of the nearly-optimal junta
tester. Our new proof replaces the Fourier machinery of the original proof with
a purely combinatorial argument that exploits the connection between sets of
variables with low influence and intersecting families.
  Another important ingredient in our proofs is a new notion of symmetric
influence. We use this measure of influence to prove that partial symmetry is
efficiently testable and also to construct an efficient sample extractor for
partially symmetric functions. We then combine the sample extractor with the
testing-by-implicit-learning approach to complete the proof that partially
symmetric functions are efficiently isomorphism-testable.
