No, twisting a firm rod or any physical object left and right would not allow for the transmission of information faster than the speed of light.
According to the theory of special relativity, information cannot travel faster than the speed of light in a vacuum. When you twist or manipulate a physical object, the information about that manipulation can only propagate through the object at the speed of sound within that material. The speed of sound in any solid material is significantly slower than the speed of light, so the information about the twisting motion would propagate at that slower speed.
In the scenario you described, if you twist one end of the 350,000 km long rod, the information about that twist would propagate through the rod at the speed of sound in that material. It would take a considerable amount of time for the twisting motion to reach the other end of the rod. This delay would be limited by the speed of sound in the material and would not exceed the speed of light.
Therefore, twisting a rod or any physical object does not provide a means to transmit information faster than the speed of light. The fundamental limitations imposed by the theory of relativity prevent such superluminal communication.