measuring the same thing at two different times, e.g., vote
at successive elections
comparing two judges' or raters' opinions
comparing the same concept measured on linked objects, e.g.,
mobility tables linking parents' class (origin) to respondents'
class (destination)
any before-and-after comparison.
The squareness often gives these tables special structure
which we can take advantage of with some specialised models.
Often the diagonal is overpopulated:
the raters agree
the respondent is in the same state before and after
the child has the same social class as the parent
Sometimes the distribution of the two variables is
(approximately) the same: marginal homogeneity may arise
e.g., if two raters are trying to get the `correct' proportions in
each category.
Sometimes there is symmetry: moves from to are as
likely as moves from to and the table is symmetrical
around the main diagonal.
However, if there is not marginal homogeneity
e.g., the second rater is more prone to grade as `good' and
`very good'
the class distribution has changed substantially while the respondent
was growing up
the government got thrown out
there cannot be true symmetry: the model that
fits `as much symmetry as can be expected' is the
quasi-symmetry model.
to 
are as
likely as moves from 
to 
and the table is symmetrical
around the main diagonal.
