Yes, torque can have a negative value in physics.
Torque is vector energy. Torque is the vector analogue of Work involving force (F) and displacement (D) vectors and the angle (FD).
For example Work W = -F.D= -|FD|cos(FD) and Toque T = FxD =|FD|sin(FD).
Torque can be negative dependent on the sine(FD).
Work and torque is an example of the quaternion nature of physics; for example
Quaternion energy E = FD = -F.D + FxD, the real energy is called work F.D and the vector energy is called torque, FxD.
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Yes Work can be positive or Negative.
If the absolute value of the negative is bigger than that of the positive, then the answer is negative. If the absolute value of the negative is the same, then zero. If the absolute value of the negative is smaller, then positive. Absolute value is the value ignoring the sign.
cross: torque dot: work
The result will be a negative value.
An absolute value can not be negative.