@Doc, I have the 1.5.x tiny cores, everything seems like the latest version. I gave up on the USBtinyISP and I went back to trying to use the Due as the ISP programmer.
I downloaded Peter VH's bitbanging ArduinoISP code and ran it as a sketch instead of the stock example ArduinoISP in the IDE.
http://forum.arduino.cc/index.php?topic=156764.0
It loads and gives an LED heartbeat. Progress.
You have to modify /hardware/arduino/avr/programmers.text:
#arduinoisp.protocol=stk500v1
arduinoisp.protocol=arduino
You have to be careful to keep all Vcc at 3V3, board and chip.
Switch over to Due Native USB port to upload sketch to ATTiNY.
More progress. I can now see SPI banging away on my newly arrived Saleae logic analyzer (an elegant tool). I have some wiring details to correct and then I am going to see if I can talk to the ATTINY in earnest.