All magnets elongate and contract anisotropically when placed in a magnetic field, an effect referred to as Joule magnetostriction. The hallmark of Joulian magnetostriction is volume conservation, which is a broader definition applicable to self-accommodation of ferromagnetic, ferroelectric or ferroelastic domains in all functional materials. Here we report the discovery of 'giant' non-volume-conserving or non-Joulian magnetostriction (NJM). Whereas Joulian strain is caused by magnetization rotation, NJM is caused by facile (low-field) reorientation of magnetoelastically and magnetostatically autarkic (self-sufficient) rigid micro-'cells', which define the adaptive structure, the origin of which is proposed to be elastic gradients ultimately caused by charge/spin density waves. The equilibrium adaptive cellular structure is responsible for long-sought non-dissipative (hysteresis-free), linearly reversible and isotropic magnetization curves along all directions within a single crystal. Recently discovered Fe-based high magnetostriction alloys with special thermal history are identified as the first members of this newly discovered magnetic class. The NJM paradigm provides consistent interpretations of seemingly confounding properties of Fe-based alloys, offers recipes to develop new highly magnetostrictive materials, and permits simultaneously large actuation in longitudinal and transverse directions without the need for stacked composites.