  A lack of correlations in the microwave background temperature between sky
directions separated by angles larger than 60 degrees has recently been
confirmed by data from the Planck satellite. This feature arises as a random
occurrence within the standard LCDM cosmological model less than 0.3 per cent
of the time, but so far no other compelling theory to explain this observation
has been proposed. Here we investigate the theoretical cross-correlation
function between microwave background temperature and the gravitational lensing
potential of the microwave background, which in contrast to the temperature
correlation function depends strongly on gravitational potential fluctuations
interior to our Hubble volume. For standard LCDM cosmology, we generate random
sky realizations of the microwave temperature and gravitational lensing,
subject to the constraint that the temperature correlation function matches
observations, and compare with random skies lacking this constraint. The
distribution of large-angle temperature-lensing correlation functions in these
two cases is different, and the two cases can be clearly distinguished in
around 40 per cent of model realizations. We present an a priori procedure for
using similar large-angle correlations between other types of data, to
determine whether the lack of large-angle correlations is a statistical fluke
or points to a shortcoming of the standard cosmological model.
