Curiously, a word referring to the handing down of beliefs and customs and one pertaining to a breakdown in fidelity to a political system, which is based on beliefs and customs, though they are not antonyms, have a common etymology. This post discusses these words and several others with the same ancestor.
The words listed below all derived from tradere, a Latin verb meaning “deliver” or “hand over.” That word, in turn, stems from a combination of the Latin preposition trans, meaning “over” (seen in words such as transfer and transport) and the Latin verb dare, meaning “do.” Interestingly, however, though to trade is to deliver or hand over (in return for something else), the English word trade is not related; its origin is the Germanic trade, meaning “course” or “track” and cognate with tread. (Likewise, the English verb and noun dare is from Old English, not Latin.)
Tradition comes from traditionem, referring to an act of delivery or handing over; the adjectival and adverbial forms are traditional and traditionally. (Trad occasionally appears as a slang abbreviation of traditional.) Adherence to tradition is called traditionalism, and one who advocates that philosophy is a traditionalist. Extradition, meanwhile, refers to handing over, as when the authorities in one country deliver a fugitive to those in the country in which he or she committed a crime; the verb is extradite.
This fugitive may very well be a traitor to the country to which he or she is being extradited. Traitor, from the Latin noun traditor by way of French, means “one who delivers,” originally in the sense of information injurious to one nation and beneficial to an antagonistic country. By extension, one who merely betrays another’s trust may be branded a traitor. The act of betrayal is called treason, and the adjectival form is treasonous (and, less often, treasonable, with the adverbial form treasonably); however, treasonously is not employed as an adverb. (Treachery and its similarly inflected adjectival and adverbial forms is a similar-looking but unrelated synonym.)
Speaking of betray, that word’s root stems from tradere as well. (An act of unfaithfulness is betrayal, and the actor is a betrayer.)