En inglés, en los caballos al igual que en los coches, "off" es la derecha y "near" es la izquierda. Es que, como conducimos a la izquierda (en el RU), la izquierda está más cerca ("near") de lado del camino o carretera.

"off
adjective
3 [attributive] Located on the side of a vehicle that is normally furthest from the curb; offside."
https://en.oxforddictionaries.com/definition/us/off#off

"near
adjective
4 [attributive] Located on the side of a vehicle that is normally closest to the curb."
https://en.oxforddictionaries.com/definition/us/near

"That off pastern angle (right one) definitely looks a little steeper than the left one. "
http://forum.cyberhorse.com.au/forums/archive/index.php/t-45...
(Esto es de Australia, donde también conducen a la izquierda.)

"When the horse leads with the right or off fore-leg, and follows with the left or near hind-leg, he is said to canter false"
https://books.google.es/books?id=xz8AAAAAQAAJ&pg=PA67&lpg=PA...

No dice si se trata de la pata derecha delantera o la trasera; a lo mejor el contexto lo aclara. Pero "off" significa "derecho/a".

Pastern, efectivamente, es cuartilla:

http://diccionario.reverso.net/ingles-espanol/pastern

"The pastern is a part of the leg of a horse between the fetlock [menudillo, tobillo] and the top of the hoof."
https://en.wikipedia.org/wiki/Pastern

"cuartilla
5. f. En las caballerías, parte que media entre los menudillos y la corona del casco."
http://dle.rae.es/?id=BTSsS6l

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Note added at 2 hrs (2017-03-01 19:48:21 GMT)
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Hablando de caballos, "off" significa la derecha también en Estados Unidos, a pesar de que conducen a la derecha, como en España. La autora del siguiente blog es estadounidense:

"Near and Off Side
The horse’s left is his near side, the right his off side. We lead horses, do up their tack, and mount from the near side. Believe it or not, this all goes back to the days of wearing swords on horseback. Since a sword is usually hung from the left hip, mounting from the left means the scabbarded sword does not have to cross the horse’s back as the rider swings his leg over. Also, if you are mounting with sword in hand, your left hand is able to control the reins, and the right hand your sword, as you mount."
http://www.academieduello.com/news-blog/off-side-navigating-...