Thanks to you both.
My Onkyo receiver has a standby state and a power off state. If it is set up to go into network standby or certain HDMI pass-though modes, it goes into the standby state when I send the Power Off command from iRule.
If the receiver loses power, or it is powered off in certain HDMI pass-though modes, it goes into the power off state.
The power on codes supplied by iRule for the receiver turn it on if it is standby but not if it is in the power off state.
The Onkyo remote control sends an IR command (POWER) that turns the receiver on when it is in the power off state. This is the IR command that I tried to learn on my pronto and that 3FG has provided. iRule sent me three IR power codes identified as: Power On, Power Off and Power Toggle. The one 3FG has provided is the Power Toggle of those three. This makes sense, because the power key on the remote operates as a toggle switch.
So there should be an end in site. Unfortunately, so I haven't been able to get that code to work on the iRule.
If SysIntegration has other codes I could try, I will do that. Other than that, I'll continue to bug the (very patient and helpful) iRule online help guy.