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Twelfth Night,
Act 5, Scene 1, lines 6-7.

Note to Twelfth Night, 5.1.6-7, "This is, to give a dog, and in recompense desire my dog again"

In other words, "This is as though I give you a dog, and then, in recompense, ask for my dog back."

The speech is perhaps an allusion to an event, or a rumor of an event, which is recorded in the diary of John Manningham, who writes:
Mr. Francis Curle told me howe one Dr. Bullein, the Queenes kinsman, had a dog which he doted one, soe much that the Queene understanding of it requested he would graunt hir one desyre, and he should have what soever he would aske. Shee demaunded his dogge; he gave it, and "nowe, Madame," quoth he, "you promised to give me my desyre." "I will," quothe she. "Then I pray you give me my dog again."1
This passage comes two pages after the Manningham's mention of the death of Queen Elizabeth on 24 March 1602. The death of the Queen seems to have reminded him of this story, so perhaps the story was a bit of popular lore associated with the Queen.

(Manningham's diary also contains a paragraph about Twelfth Night, which he saw on 2 February 1601.)

     1John Manningham, Diary of John Manningham, of the Middle Temple, and of Bradbourne, Kent, Barrister-at-Law, 1602-1603, ed. John Bruce (1868; New York: Johnson Reprint Corporation, 1968) 148-49.

Queen Elizabeth I with dog