  We introduce and study the notion of null-orbit reflexivity, which is a
slight perturbation of the notion of orbit-reflexivity. Positive results for
orbit reflexivity and the recent notion of $\mathbb{C}$-orbit reflexivity both
extend to null-orbit reflexivity. Of the two known examples of operators that
are not orbit-reflexive, one is null-orbit reflexive and the other is not. The
class of null-orbit reflexive operators includes the classes of hyponormal,
algebraic, compact, strictly block-upper (lower) triangular operators, and
operators whose spectral radius is not 1. We also prove that every polynomially
bounded operator on a Hilbert space is both orbit-reflexive and null-orbit
reflexive.
