  The violent collisions of black holes provide for excellent test-beds of
Einstein's general relativity in the strong/dynamical gravity regime. We here
demonstrate the resolving power one can gain upon the use of multi-band
observations of gravitational waves from both ground- and space-based
detectors. We find significant improvement in both generic parameterized tests
of general relativity and consistency tests of inspiral-merger-ringdown parts
of the waveform over single-band detections. Such multi-band observations are
crucial for unprecedented probes of e.g. parity-violation in gravity.
