They all mention that when a signal enters the grid of a triode, it exits the plate ...
EDIT: Sluckey said it shorter, typed faster...

Are you ready to
really understand what is happening? If yes, then do yourself a favor and delete from your brain the part about "a signal enters ... a signal exits." We will not get far thinking about a tube in this manner, even though it is often said that way in simplistic explanations.
If you heat a tube's cathode and apply a large positive voltage to the plate, a large current flows between the plate and cathode; this is a diode (like a rectifier tube).
If you want to control the current flow with a steady voltage applied to the plate, you can insert a grid between the cathode and plate (usually closer to the cathode). This makes a triode, and a small voltage (relative to the plate) on the grid is much more effect at controlling the current than a change of plate voltage (the ratio of the two causing equal changes is the tube's Mu, or amplification factor).
All voltages we're talking about are relative to the tube's cathode. If we say "grid voltage change" it is really a change in the voltage measured from cathode to grid; if we say "plate voltage" it's really the measured voltage from cathode to plate.
Amplification ExamplePretend we have a 12__7 triode stage with a 100kΩ plate resistor, a supply voltage of 300vdc and a grid voltage of -2v which causes an idle current of 1mA. Let's also assume there is no cathode resistor; the cathode is grounded and is at 0vdc. Ohm's Law says that the 1mA through the 100kΩ plate load resistor causes a
voltage drop of 1mA * 100kΩ = 100v across the resistor, leaving 300v - 100v = 200vdc on the 12__7 plate.
Now let's say the grid voltage changes from -2v to -1v, and that change (+1v) causes the tube to pass 1.5mA. What is the voltage drop across the plate load resistor? 1.5mA * 100kΩ = 150v, which leaves 300v - 150v = 150vdc at the plate of the tube. So a 1v increase of grid voltage caused the tube to pass a current which created a -50v shift in plate voltage due to the voltage drop across the plate load resistor (from 200vdc to 150vdc).
Let's continue by saying the grid voltage changes from -2v to -3v, and that change (-1v) causes the tube to pass 0.5mA. What is the voltage drop across the plate load resistor? 0.5mA * 100kΩ = 50v, which leaves 300v - 50v = 250vdc at the plate of the tube. So a 1v decrease of grid voltage caused the tube to pass a current which created a +50v shift in plate voltage due to the voltage drop across the plate load resistor (from 200vdc to 250vdc).
- At all times, a tube voltage change which causes an increase in tube current also causes an increased voltage drop through resistances connected to the tube. Because the plate load resistor is connected from a fixed supply voltage to the plate, an increased voltage drop causes a decrease in plate voltage.
This is the reason for the apparent polarity change. - Consider the same increase in tube current causing an increased voltage drop through a resistance connected between the tube's cathode and ground. Because the cathode resistor is connected from the 0vdc reference point (ground), an increased voltage drop causes an increase in cathode voltage. A rising grid voltage would have caused the increase in tube current causing this increased voltage drop.
This is the reason for the apparent same-polarity output at the cathode.So there is no magic inside the tube, except in the manner in which a grid can allow a small applied voltage to control a relatively large current through the tube. Everything else we observe (voltage gain, signal polarity) is really just a byproduct of the interaction of circuit resistances with that tube current.