In FotR I, ch. XII, Glorfindel mentions his saddle:
'You shall ride my horse,' said Glorfindel. 'I will shorten the stirrups up to the saddle-skins, and you must sit as tight as you can.
Later in the novel, it is said at least twice that I have found with a quick ctrl + F
that the Elves ride bareback, including when Gandalf calls it "elf-fashion" (bk. III, Ch. XI).
Why, then, does Glorfindel use a saddle?