The John Adams "Mirror-Image" Portrait
Supporting the hypothesis that Thomas Jefferson may have given the Delapierre canvas to John and Abigail Adams is that in 1788 the American artist Mather Brown, working in London, painted—at Jefferson's request—a life portrait of John Adams1 2 that mirrors in some respects the 1785 Delapierre portrait (see side-by-side comparison above). In particular, the placement and angle of the quill, pages, and other items in the Adams portrait appear to mirror the placement and angle of corresponding items in the Delapierre portrait.
Unless the compositional parallels in these two portraits are mere coincidences, it appears that Brown would have needed access to the 1785 Delapierre portrait in London when he painted the Adams portrait there in 1788.
Jefferson visited the Adamses in London in the spring of 1786 and could have given the Delapierre portrait to them at that time.3 If he did, then it is likely that this was the portrait of Jefferson mentioned by Abigail Adams in her letters to him dated 23 July 1786 and 26 June 1787.
References and notes
 Jefferson, in a letter from Paris dated 22 October 1786, requested "Mr. Adams to set for his picture" and for "Mr. Brown to draw it for me." (The Papers of Thomas Jefferson, Volume 10, edited by Julian P. Boyd, Princeton University Press, Princeton, New Jersey, 1954, p. 479.) This portrait was delivered to Jefferson in Paris on 10 September 1788. (The Papers of Thomas Jefferson, Volume 13, edited by Julian P. Boyd, Princeton University Press, Princeton, New Jersey, 1956, p. 597.) It features a book labeled "Jefferson's Hist. of Virginia"—a notional depiction of Jefferson's Notes on the State of Virginia, the only full-length book he published during his lifetime (see full text of the book at the University of Virginia Library website). The Adams portrait was sold by Jefferson's heirs on 19 July 1833 at a Harding Gallery auction in Boston to an unidentified buyer, descended to George Francis Parkman, and was bequeathed by him to the Boston Athenæum in 1908. (Marc Leepson, Saving Monticello, The Free Press, Simon & Schuster, New York, 2001, p. 16.) But the gift was part of a diverse collection of items from Parkman, and neither the name of the artist nor that of the subject was known when the bequest was made. According to Dr. David B. Dearinger, the Susan Morse Hilles Curator of Paintings and Sculpture at the Boston Athenæum: "Shortly after acquiring the portrait, the Athenæum made a systematic appeal to a number of American and European scholars for their assistance with the problem of attribution. When that attempt failed, the possibility of selling the portrait for four hundred dollars was raised in 1913. Fortunately, no action was taken; in 1917, however, historian and antiquarian Lawrence Park convincingly argued that the painting was Mather Brown's long-lost portrait of John Adams." (For more details and sources, see http://www.bostonathenaeum.org/node/983)
 Susan R. Stein, The Worlds of Thomas Jefferson at Monticello, Harry N. Abrams, Inc., New York, 1993, pp. 126-127.
 For a detailed timeline of events during and after Jefferson's trip to London, and for additional evidence that Jefferson may have given the Delapierre painting to John and Abigail Adams there in the spring of 1786, see Evidence for an Early Unidentified Portrait of Thomas Jefferson.