Double acrostic
A double acrostic is a puzzle in which a set of clued answers spells out one hidden message down their first letters and a second hidden message down their last letters.
It was a Victorian parlour favourite, and the two hidden messages are usually connected — a town and its river, a writer and a book. Because each answer is pinned at both ends, a solver who has one message half-built can often force the answers that complete the other, which is the pleasure of the form.
The modern American puzzle that magazines simply call an acrostic is a descendant. Elizabeth Kingsley published the first in the Saturday Review in 1934 under the name double-crostic. You answer a numbered list of clues, and the initials of those answers spell the author and title of a quotation, while the letters themselves transfer by number into a grid that spells the quotation out.
That transfer runs both ways, which is what makes the puzzle solvable when the clues are hard. A guessed word in the quotation sends letters back into the clue answers, and a solved clue sends letters forward into the quotation.