I am intrigued by your idea of a fully symmetrical AC switch -- how would you keep such a device off, and then later how would you turn it on?
Most of the structure of a triac is fairly symmetrical.
However, as the schematic diagram tries to imply, the gate is attached "closer" to MT1.
With the triac off, a typical circuit drives the voltage at MT2 up and down hundreds of volts relative to MT1, but as long as the gate voltage stays the same as the MT1 voltage, the triac stays off.
One way to turn the triac on is to "pull down" the gate.
The circuit is the same as with a PNP transistor -- the "+5V" VCC of the microcontroller is connected to MT1; a gate resistor connects a microcontroller output pin to the gate of the triac; and the motor or other load is connected to MT2.
As Thomas O mentioned, another way to turn a triac on is to connect the gate resistor to a voltage that pulls away from the MT1 voltage in the same direction as the voltage at MT2 -- positive voltage anytime during the half-cycle when MT2 is positive, negative voltage anytime during the half-cycle when MT2 is negative.
Recently I blew up over a dozen triacs learning that (a) once a triac is on, turning a triac off is a bit more difficult, and (b) hooking the the "GND" pin of the microcontroller to MT1 and pulling the gate through a resistor to +5 V doesn't work right with a logic level triac.
The rough draft of the "Power Electronics" book already links to some schematic diagrams illustrating how to use triacs, and links to "Thyristors & Triacs - Ten Golden Rules for success in your application". Perhaps you will find those ten rules useful, and perhaps you can help us make that Wikibook better.
NXP has a good explanation as to what the differences between 3-quadrant and 4-quadrant triacs are.
Summary: a 3-quadrant triac cannot be turned on in the fourth quadrant (T2 negative, G positive) which is beneficial when dealing with inductive loads - ringing caused by sudden turn-off could potentially turn on a 4-quadrant triac, necessitating snubbers and other protection means to prevent spurious conduction.
Best Answer
When the current across the triac falls under \$I_H\$, which is the holding current, the triac stops conducting. With a pure resistive load this happens at the very end of the sine wave cycle, and voltage and current are in phase. When the load has an inductive component (e.g. a motor) then there is a lag between current and voltage. At the moment when the current drops below \$I_H\$, the voltage has already risen with the opposite polarity. Therefore when the triac turns off there is a big dV/dt on the triac - "voltage is cut off immediately". This situation can lead to self triggering the triac, and it starts to conduct uncontrolled. The remedy is to use a snubber circuit, i.e. an RC in parallel with the triac.