It doesn’t matter whether the bride or the bridegroom writes the letters of thanks for wedding presents provided that these go out immediately after the arrival of each present and are not in the handwriting of the bride’s mother.

Judith Martin (b. 1938) American author, journalist, etiquette expert [a.k.a. Miss Manners]

Miss Manners on Painfully Proper Weddings, ch. 13 “Giving and Receiving Thanks” (1995)
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Also titled in later editions, Miss Manners on Weddings and Miss Manners' Guide to a Surprisingly Dignified Wedding.