The common phrase in both is the bending in of the knees on the exhale while
arms reach out past the hips.
My training always had the back stroke reach end with palms down over thighs,
but reaching past the legs sets up the same expression as coordination, where
the knees come in against the arms reaching out.
In coordination the arms lever at the elbow, in backstroke the lever increases
out to the shoulder.
In coordination the arms flex and extend at the elbow while the legs open
and close in the lateral plane.
In backstroke arms and legs rotate through synchronous planes. (rotation following
in-plane patterns of movement)
What seems to be the most common pattern of breathing has the arms and legs
moving through the reach on an exhale.
If you inhale out into the reach (just like on the coordination), you have the
exhale to bend your knees through the reach of the arms.
It's a beautiful thing when the two exercises, coordination and backstroke,
express the same energy.