There's no such phraseology as "downwind takeoff." The controller is
required to issue the wind when you takeoff with a tailwind component,
this can be accomplished via the ATIS and would be obvious when you
listened to the tape.

Jim Macklin wrote:

If a tower phrased a take-off clearance "cleared for |
downwind takeoff" it would be because the pilot requested
that runway and the tower was getting it on tape, that the
pilot knew and accepted the procedure. If the wind was not
significant and a taxi to the other end of the airport was
long, using a runway with a little wind component on the
tail might be very reasonable. It depends on runway length,
departure profile and terrain balanced by aircraft
performance.