  We cross-correlate the largest available Mid-Infrared (WISE), X-ray (3XMM)
and Radio (FIRST+NVSS) catalogues to define the MIXR sample of AGN and
star-forming galaxies. We pre-classify the sources based on their positions on
the WISE colour/colour plot, showing that the MIXR triple selection is
extremely effective to diagnose the star formation and AGN activity of
individual populations, even on a flux/magnitude basis, extending the
diagnostics to objects with luminosities and redshifts from SDSS DR12. We
recover the radio/mid-IR star formation correlation with great accuracy, and
use it to classify our sources, based on their activity, as radio-loud and
radio-quiet AGN, LERGs/LINERs, and non-AGN galaxies. These diagnostics can
prove extremely useful for large AGN and galaxy samples, and help develop ways
to efficiently triage sources when data from the next generation of instruments
becomes available. We study bias in detail, and show that while the widely-used
WISE colour selections for AGN are very successful at cleanly selecting samples
of luminous AGN, they miss or misclassify a substantial fraction of AGN at
lower luminosities and/or higher redshifts. MIXR also allows us to test the
relation between radiative and kinetic (jet) power in radio-loud AGN, for which
a tight correlation is expected due to a mutual dependence on accretion. Our
results highlight that long-term AGN variability, jet regulation, and other
factors affecting the $Q/L$$_{bol}$ relation, are introducing a vast amount of
scatter in this relation, with dramatic potential consequences on our current
understanding of AGN feedback and its effect on star formation.
