score:27
You mentioned Calhoun - it seems he definitely did cast two tie-breaking votes (on the same subject, but separate occasions) against the wishes of the President. This was on Andrew Jackson's nomination of Martin Van Buren as ambassador to the UK; it seems to have been quite a complicated intraparty struggle, with Calhoun trying to prevent Van Buren becoming Jackson's successor.
Twice, however, in the hands of Vice-President Calhoun the casting ballot served effectively as a weapon against President Jackson, for Callhoun was able first (January 13, 1832) to halt the nomination of Martin Van Buren as minister to England and finally (January 25) to defeat it. The incidents of these votes were of peculiar interest inasmuch as Van Buren was at the time in London acting as our minister on a recess commission, and his rejection helped to make him Vice-President. Moreover it is altogether probable that Calhoun's partizans in the Senate provided him with the opportunity which he sought thus to assert his spite against Jackson.
(WP clarifies that last comment - enough of Calhoun's faction abstained to make it into a tie, which he was then able to break. So it seems to have been a bit of a manufactured situation.)
None of the other cases mentioned by Learned are explicitly described as against Presidential policy, and I'm not familiar enough with the political nuance to say for sure either way. It also doesn't cover any cases since 1915 (obviously). So I can't say for sure Calhoun was the only one, but it seems quite possible.