Doff your hat
Raise your hat in acknowledgement of or deference to another.
Doffing seems to be an activity that is limited to hats or caps; it isn't often that we hear the verb used in any other connection. That needn't be the case though, as becomes apparent when one realises the connection between 'doffing' and 'donning'. We can don hats and caps, but we can also also don clothes, shoes, even a persona or a set of ideas and, as doffing is the opposite of donning, anything that can be donned can later be doffed. The origins of these strange little words are simply 'do on' and 'do off'.
When the terms 'doff' and 'don' were first used there was no especial connection with headgear. The first usage of it that I can find in print is from Sir Thomas Malory's Le morte Darthur, circa 1470:
Doffe of thy clothes, And knele in thy kyrtylle. [A tunic or petticoat]
Shakespeare was fond of the word 'doff' and used it frequently, often in a figurative manner, which alludes both to the removal of clothes and of opinions. In King John, 1595, he has Constance say "Thou weare a Lyons hide! doff it for shame." and in The Taming of the Shrew, 1596 "Fie, doff this habit, shame to your estate". He also frequently used the alternative 'daff' and its past tense 'daft', as in the 1597 sonnet A Lover's Complaint: "There my white stole of chastity I daft."
The falling out of daily use of 'doff' isn't just because men no longer routinely wear hats - the usage appears to be geographically biased. Here in the North of England, caps are still doffed, whereas in Scotland the term was considered archaic even by the 18th century. Samuel Johnson defined 'doff' in A dictionary of the English language, 1755, as "to put off dress; to strip" but later dismissed it as "in all its senses obsolete, and scarcely used except by rustics". In America, gentlemen have always preferred to 'tip' their hats, that is, signal a salutation by a slight tug at the hat's rim, rather than to doff them, which involves a brief removal.
In the 16th to the 18th centuries in England, the donning and doffing of hats was governed by a code of etiquette and custom that it is hard for us now to appreciate. Every man of standing wore a hat, and the form of hat and the rules governing when it could be removed or for whom it should be raised in acknowledgement were bewilderingly complex. Hat doffing was an accompaniment to bowing and the depth of the bow determined how far the hat was lifted. In 1896, James Boyle, in an attempt to remove the drudgery of continually lifting his hat, registered a patent for a form of self-doffing device. When the wearer bowed to an acquaintance, the hat lifted itself, rotated once and lowered again. The hats didn't catch on. As Shakespeare might have put it - it was a daft idea.