  We search NANOGrav's 12.5-year data set for evidence of a gravitational wave
background (GWB) with all the spatial correlations allowed by general metric
theories of gravity. We find no substantial evidence in favor of the existence
of such correlations in our data. We find that scalar-transverse (ST)
correlations yield signal-to-noise ratios and Bayes factors that are higher
than quadrupolar (tensor transverse, TT) correlations. Specifically, we find ST
correlations with a signal-to-noise ratio of 2.8 that are preferred over TT
correlations (Hellings and Downs correlations) with Bayesian odds of about
20:1. However, the significance of ST correlations is reduced dramatically when
we include modeling of the Solar System ephemeris systematics and/or remove
pulsar J0030$+$0451 entirely from consideration. Even taking the nominal
signal-to-noise ratios at face value, analyses of simulated data sets show that
such values are not extremely unlikely to be observed in cases where only the
usual TT modes are present in the GWB. In the absence of a detection of any
polarization mode of gravity, we place upper limits on their amplitudes for a
spectral index of $\gamma = 5$ and a reference frequency of $f_\text{yr} = 1
\text{yr}^{-1}$. Among the upper limits for eight general families of metric
theories of gravity, we find the values of $A^{95\%}_{TT} = (9.7 \pm 0.4)\times
10^{-16}$ and $A^{95\%}_{ST} = (1.4 \pm 0.03)\times 10^{-15}$ for the family of
metric spacetime theories that contain both TT and ST modes.
