Cosmological models with Galileon gravity are an alternative to the standard
$\Lambda {\rm CDM}$ paradigm with testable predictions at the level of its
self-accelerating solutions for the expansion history, as well as large-scale
structure formation. Here, we place constraints on the full parameter space of
these models using data from the cosmic microwave background (CMB) (including
lensing), baryonic acoustic oscillations (BAO) and the Integrated Sachs-Wolfe
(ISW) effect. We pay special attention to the ISW effect for which we use the
cross-spectra, $C_\ell^{\rm T g}$, of CMB temperature maps and foreground
galaxies from the WISE survey. The sign of $C_\ell^{\rm T g}$ is set by the
time evolution of the lensing potential in the redshift range of the galaxy
sample: it is positive if the potential decays (like in $\Lambda {\rm CDM}$),
negative if it deepens. We constrain three subsets of Galileon gravity
separately known as the Cubic, Quartic and Quintic Galileons. The cubic
Galileon model predicts a negative $C_\ell^{\rm T g}$ and exhibits a
$7.8\sigma$ tension with the data, which effectively rules it out. For the
quartic and quintic models the ISW data also rule out a significant region of
the parameter space but permit regions where the goodness-of-fit is comparable
to $\Lambda {\rm CDM}$. The data prefers a non zero sum of the neutrino masses
($\sum m_\nu\approx 0.5$eV) with $ \sim \! 5\sigma$ significance in these
models. The best-fitting models have values of $H_0$ consistent with local
determinations, thereby avoiding the tension that exists in $\Lambda {\rm
CDM}$. We also identify and discuss a $\sim \! 2\sigma$ tension that Galileon
gravity exhibits with recent BAO measurements. Our analysis shows overall that
Galileon cosmologies cannot be ruled out by current data but future lensing,
BAO and ISW data hold strong potential to do so.