  We perform a joint analysis of the power spectrum and the bispectrum of the
CMB temperature and polarization anisotropies to improve the constraints on
isocurvature modes. We construct joint likelihoods, both for the existing
Planck data, and to make forecasts for the future LiteBIRD and CMB-S4
experiments. We assume a general two-field inflation model with five free
parameters, leading to one isocurvature mode (which can be CDM density,
neutrino density or neutrino velocity) arbitrarily correlated with the
adiabatic mode. We theoretically assess in which cases (of detecting and/or
fixing parameters) improvements can be expected, to guide our subsequent
numerical analyses. We find that for Planck, which detected neither
isocurvature modes nor primordial non-Gaussianity, the joint analysis does not
improve the constraints in the general case. However, if we fix additional
parameters in the model, the improvements can be highly significant depending
on the chosen parameter values. For LiteBIRD+CMB-S4 we study in which regions
of parameter space compatible with the Planck results the joint analysis will
improve the constraints or the significance of a detection. We find that, while
for CDM isocurvature this region is very small, for the neutrino isocurvature
modes it is much larger. In particular for neutrino velocity it can be about
half of the Planck-allowed region, where the joint analysis reduces the
isocurvature error bars by up to 70%. In addition the joint analysis can also
improve the error bars of some of the standard cosmological parameters, by up
to 30% for $\theta_{MC}$ for example, by breaking the degeneracies with the
correlation parameter between adiabatic and isocurvature modes.
