This statement is certainly false:

* If two items are equal, and pairwise inequality is deterministic, exchanging the items does not affect the sorting of other items in the list.

Just to demonstrate this obviousness:

>>> sorted([9, 9, 9, b, 1, 2, 3, a])
[1, 2, 3, A, B, 9, 9, 9]
>>> sorted([9, 9, 9, a, 1, 2, 3, b])
[B, 9, 9, 9, A, 1, 2, 3]
>>> a == b

The classes involved are:

class A:
    def __lt__(self, other):
        return False
    __gt__ = __lt__
    def __eq__(self, other):
        return True
    def __repr__(self):
        return self.__class__.__name__

class B(A):
    def __lt__(self, other):
        return True
    __gt__ = __lt__

I do not think these are useful, but __lt__ is deterministic here.

Keeping medicines from the bloodstreams of the sick; food
from the bellies of the hungry; books from the hands of the
uneducated; technology from the underdeveloped; and putting
advocates of freedom in prisons.  Intellectual property is
to the 21st century what the slave trade was to the 16th.