A nonlinear system possesses an invariance with respect to a set of transformations if its output dynamics remain invariant when transforming the input, and adjusting the initial condition accordingly. Most research has focused on invariances with respect to time-independent pointwise transformations like translational-invariance (u(t) bar right arrow u(t) p, p is an element of R) or scale-invariance (u(t) bar right arrow pu(t), p is an element of R->0). In this article, we introduce the concept of s(o)-invariances with respect to continuous input transformations exponentially growing/decaying over time. We show that so-invariant systems not only encompass linear time-invariant (LTI) systems with transfer functions having an irreducible zero at so E R, but also that the input/output relationship of nonlinear s(o)-invariant systems possesses properties well known from their linear counterparts. Furthermore, we extend the concept of s(o)-invariances to second-and higher-order soinvariances, corresponding to invariances with respect to transformations of the time-derivatives of the input, and encompassing LTI systems with zeros of multiplicity two or higher. Finally, we show that nth order 0-invariant systems realize - under mild conditions - nth-order nonlinear differential operators: when excited by an input of a characteristic functional form, the system's output converges to a constant value only depending on the nth (nonlinear) derivative of the input. (C) 2017 The Authors. Published by Elsevier Ltd.