And the hare, because she cheweth the cud but parteth not the hoof, she is unclean unto you. (ASV)
Hares, more usually called rabbits nowadays, do not chew the cud.
Actually, they do - it's a process called "cecotrophy", and while it's not the exact same as the way a cow does it, it's essentially rumination (in fact, another word for cecotrophy is "pseudo-rumination").
From Keil & Delitzsch Commentary on the Old Testament
"The hare and hyrax (Klippdachs) were also unclean, because, although they ruminate, they have not cloven hoofs. It is true that modern naturalists affirm that the two latter do not ruminate at all, as they have not the four stomachs that are common to ruminant animals; but they move the jaw sometimes in a manner which looks like ruminating, so that even Linnaeus affirmed that the hare chewed the cud, and Moses followed the popular opinion. According to Bochart, Oedmann, and others, the shaphan is the jerboa, and according to the Rabbins and Luther, the rabbit or coney. But the more correct view is, that it is the wabr of the Arabs, which is still called tsofun in Southern Arabia (hyrax Syriacus), an animal which feeds on plants, a native of the countries of the Lebanon and Jordan, also of Arabia and Africa. They live in the natural caves and clefts of the rocks (Ps 104:18), are very gregarious, being often seen seated in troops before the openings to their caves, and extremely timid as they are quite defenceless (Prov 30:26). They are about the size of rabbits, of a brownish grey or brownish yellow colour, but white under the belly; they have bright eyes, round ears, and no tail. The Arabs eat them, but do not place them before their guests."
What the "Con" section tries to say is that hares/rabbits occasionally eat their own faeces, which happens to also be called "refection".
Structurally this can be considered to be equivalent to rumination, though the food has to leave the body before being chewed again.
The web-page Do the badger and rabbit chew cud? suggests that the badger (Leviticus 11:5) and the rabbit were classified with more true ruminants (according to modern taxonomy), because they move their jaws in the same manner as the ruminants listed.
The Hebrew word translated hare/rabbit is "arnabeth", which apparently means a "leaper"; see Leviticus 11 in the Holy Bible in Modern English by Ferrar Fenton.
For some discussion of the issue, see e.g. Chew on This... Again!.