Indeed, the airfoils on many horizontal tail surfaces do have negative camber. This is mostly in combination with powerful flaps on the wing so the tailplane will continue to work with flaps extended, when it needs to develop a relatively high downforce (which is indeed the same as negative lift). The extended wing flaps result in higher wing downwash so the tail "sees" a more negative angle of attack. By cambering the tail airfoil, it will tolerate more negative angles of attack and its minimum lift coefficient is lower than that of a symmetrical airfoil.
I know it is hard to see, but this Do-228 tailplane really does have negative camber (picture source).
The PZL Wilga has a symmetrical tail airfoil but uses an inverted, fixed slat at the leading edge of its elevator (picture source).
The A380 uses negative camber at the root of the tail, too, like most airliners (picture source). To keep isobars aligned with chord lines in a swept wing is also a reason for negative camber at the root.