does there exist type of transistor that can use both negative and positive gate voltage?
Yes.
Although your question is not very clear.
Most RF HEMTs have a Schottky barrier gate.
When the gate is at zero volts (Vgs=0), the drain current is Idss.
As the gate voltage goes negative, the drain current drops and eventually pinches off.
For positive gate voltage the drain current increases.
For higher gate voltage, the Schottky barrier gate will start to turn on and gate current will increase. Usually this is the limit of positive gate voltage. At this point the drain current goes to Imax.
So the drain current goes from zero at pinchoff to Imax.
The pinchoff voltage can be negative for a depletion mode HEMT or positive for an enhancement mode HEMT or it can be set by the processing to anything in between.
For example, a HEMT with a pinchoff voltage of -0.5V can operate with a gate voltage range of approximately -0.5V to +0.5V.
I would think that a MOSFET could be made similarly. That is, the Vt of a MOSFET can be controlled to be either positive or negative (enhancement or depletion mode). So if the threshold voltage was set to say -1V, then at zero volts it would be partially turned on, and at +1V fully turned on for example.