Do symmetric BJTs exist?
Check out "muting transistors" in consumer audio gear:

http://pdf.datasheetcatalog.com/datasheet/toshiba/978.pdf

The purpose of these is to short the outputs of a device to ground, for example while it powers up and down, to avoid producing a THUMP. Since the signal to short out is AC and has a DC component of 0V, a bidirectional switch is needed. MOSFETs would not work because the body diode would chop off half the signal when the switch is open, so the usual suspect here is a BJT, 2SC2878. Q801/Q802 here:



These are optimized for high reverse hFe and high Vebo (so the base junction doesn't act as a zener when they're off).

Apart from these special perks which make them pretty much the only BJT able to fulfill this role, the rest of their characteristics is very unremarkable.