  Modified gravity theories on cosmic scales have three key deviations from
general relativity. They can cause cosmic acceleration without a physical,
highly negative pressure fluid, can cause a gravitational slip between the two
metric potentials, and can cause gravitational waves to propagate differently,
e.g. with a speed different from the speed of light. We examine whether the
deviations in the metric potentials as observable through modified Poisson
equations for scalar density perturbations are related to or independent from
deviations in the tensor gravitational waves. We show analytically they are
independent instantaneously in covariant Galileon gravity -- e.g. at some time
one of them can have the general relativity value while the other deviates --
though related globally -- if one deviates over a finite period, the other at
some point shows a deviation. We present expressions for the early time and
late time de Sitter limits, and numerically illustrate their full evolution.
This in(ter)dependence of the scalar and tensor deviations highlights
complementarity between cosmic structure surveys and future gravitational wave
measurements.
