  We analyse partitions of products with two ordered factors in two classes
where both factors are countable or well-ordered and at least one of them is
countable. This relates the partition properties of these products to cardinal
characteristics of the continuum. We build on work by Erd\H{o}s, Garti, Jones,
Orr, Rado, Shelah and Szemer\'edi. In particular, we show that a theorem of
Jones extends from the natural numbers to the rational ones but consistently
extends only to three further equimorphism classes of countable orderings. This
is made possible by applying a thirteen-year old theorem of Orr about embedding
a given order into a sum of finite orders indexed over the given order.
