The characters for Kaenbyou literally mean "blazing cat". The one for Rin means "phosphorus".
Kasha are a type of youkai that appear amidst stormy weather to bear the bodies of sinners to Hell. They are said to be cats that have turned into youkai (e.g. through long life). Sometimes temples near places where kasha are said to live will hold two funerals for the dead: first, the coffin is filled with rocks so that the kasha won't disrupt the funeral to carry off the deceased.
In Buddhism, a kasha is a fiery chariot which carries dead sinners to hell.
Rin is shown pushing a wheelbarrow, which is also known in Japanese as a 猫車 (or literally, cat cart).
Some of her attacks resemble attacks the second boss of Samidare, that Anareta uses.
Rin may be the most persistent Touhou boss, appearing as a mid-boss in three stages in addition to being a stage boss all in one game.
Rin is the only Windows Stage 6 midboss that, when defeated, doesn't directly lead to the final boss. A generic "death fairy" leads to Utsuho instead, similar to MoF's Stage 6, except there is no midboss.
She's also the only Windows Stage 5 boss who isn't a servant or subordinate of the Stage 6 boss, since she's a pet of Satori, who is the Stage 4 boss.
In ancient Aztec culture, each household had a small red dog called the Techihi who was said to guide lost souls to the underworld, and would often be sacrificed along with its master. Strangely, Rin has both red hair and performs a similar duty.
She's who actually causes the "incident" in Subterranean Animism. (There were lots of evil spirits going around in surface).
Rin is commonly referred to as just "Orin" instead of her full name because that's what shows as her boss title in SA. The name 「おりんりん」 or "OrinRin," has also been a standard fanon way of referring to her.
Careful observers will notice that Rin has both human ears and cat ears. This has led to some fanart where she takes her cat ears off. Or in a few rare cases, the other way around.
In Double Spoiler, when taking a picture of Rin in her cat form, the player is awarded the "Cat bonus", which gives a bonus of 666 points, the mark of the beast. This is somewhat fitting considering that she is located in a Hell.