  Gravitational waves (GWs) induced by scalar curvature fluctuations are an
important source of the cosmological GW background and a crucial counterpart of
the primordial black hole scenario. However, doubts have been cast on the
theoretically predicted induced GW spectrum due to its seeming gauge
dependence. In this paper, we shed light on the gauge dependence issue of the
induced GW spectrum in general cosmological backgrounds. First, inspired by the
Hamiltonian formalism we provide very simple formulas for the tensor modes at
second order in cosmological perturbation theory. We also emphasize the
difference between observable and gauge invariant variables. Second, we argue
that the Newton (or shear-free) gauge is suitable for both the calculation of
induced GWs and the physical interpretation. We then show that, most notably,
the induced GW spectrum is invariant under a set of reasonable gauge
transformations, i.e. physically well behaved on small scales, once the source
term has become inactive. This includes the commonly used flat, constant Hubble
and synchronous gauges but excludes the comoving slicing gauge. We also show
that a particular solution of the GW equation in a dust dominated universe
while the source term is active can be gauged away by a small change of gauge.
