Palindromes are one of the oldest forms of wordplay, so it’s not surprising that there are several examples in Latin. Here are some of the most notable ones.
In girum imus nocte et consumimur igni. (“We wander at night and are consumed by fire.” Here, “wander” translates an idiom literally meaning “go in a circle.” In my opinion, this is one of the most natural palindromes in any language, and its meaning is wonderfully evocative.)
Sator Arepo tenet opera rotas. (By far the most famous Latin palindrome, it’s commonly translated as “The sower Arepo holds the wheels at work,” though its true claim to fame is that the words also form a word square. Unfortunately, opera doesn’t seem to have a place here, being in the nominative or accusative plural, but the real Achilles heel is the completely invented name Arepo. An actual proper name would be fine, but all sources agree that the “name” Arepo occurs nowhere except this palindrome and its word square.)
Subi dura a rudibus. (“Endure hardships from the rude.” A good reminder for patience, and grammatically sound.)
Si bene te tua laus taxat, sua laute tenebis. (One Latin forum credibly translates this as “If your praise rates you well, you’ll maintain its affairs splendidly,” though it’s hard to construct a context in which the sentence would make much sense.)
Roma tibi subito motibus ibit amor. (This is usually translated as “In Rome, love will come to you suddenly,” but other translations are possible, and “come” is actually “go.” The most graceful way to incorporate motibus is a matter of opinion.)
Sole medere, pede ede, perede melos. (The same Latin forum has the seemingly flawed translation “Heal with the sun, get appetite by walking, and publish your works.” The only way I can interpret melos here is as the accusative singular of “hymn,” so this would seem to say “Heal [yourself] by the sun, bring forth with [your] foot, consume a hymn.” The imperative ede could mean “bring forth,” “declare,” or “appoint [a time],” none of which is typically done with a foot, so unfortunately the meaning there is as inscrutable as consuming a hymn.)