  Minkowski Tensors are tensorial generalizations of the scalar Minkowski
Functionals. Due to their tensorial nature they contain additional
morphological information of structures, in particular about shape and
alignment, in comparison to the scalar Minkowski functionals. They have
recently been used [29] to study the statistical isotropy of temperature and E
mode data from the Planck satellite. The calculation in [29] relied on
stereographic projection of the fields to extract the shape and alignment
information. In this work, we calculate Minkowski Tensors directly on the
sphere and compute the net alignment in the data, based on a recent work that
extends the definition of Minkowski Tensors to random fields on curved spaces.
This method circumvents numerical errors that can be introduced by the
stereographic projection. We compare the resulting net alignment parameter
values obtained from the frequency coadded CMB temperature data cleaned by the
SMICA pipeline, to those obtained from simulations that include instrumental
beam effects and residual foreground and noise. We find very good agreement
between the two within $\approx$ 1$\sigma$ . We further compare the alignments
obtained from the beam-convolved CMB maps at individual Planck frequencies to
those in the corresponding simulations. We find no significant difference
between observed data and simulations across all Planck frequencies, except for
the 30 GHz channel. For the 30 GHz channel we find $\approx$ 2$\sigma$
difference between the data and the simulations. This mild disagreement most
likely originates from inaccurate estimation of the instrumental beam at 30
GHz.
