What class of audio amplifier is this unusual configuration?
It runs in class A as both output devices conduct for the entire cycle of the signal.

It can provide twice the quiescent current into the load because as Q3 increases in current, Q2 reduces. The opposite occurs for negative going signals at the input.

The quiescent current is mainly defined by the value of R4 and the gain of the transistors.  Approximately half of the current through R4 goes into the base of Q2, the other half flows through Q1 into the base of Q3 with some getting diverted into R5. 

The quiescent current will vary with the gain of the transistors although R5 and R6 will help stabilize it.

That is not a very good version - it has no bootstrapping so will be limited in output excursion and it depends upon the Hfe of the two output transistors being matched as well as having to be adjusted for the specific transistors.

One of the earliest well-known implementations of this configuration for audio was in Wireless World magazine in 1969.

Note that compared to the poster's circuit that the resistor feeding the collector of the driver transistor is split into 2 with a capacitor driving the junction from the output.  This bootstrapping allows the output to swing to the positive rail. In the poster's circuit the current through R4 reduces as the output voltage goes positive starving the output transistor of base current.

In non-audio use a similar configuration is used as the output of TTL gates.



From "http://www.angelfire.com/sd/paulkemble/sound3b.html"

Magazine article