DB, no, absolutely don’t do that. In almost all circumstances the outputs of two power amplifiers should not be connected together. You would likely damage both amps, eventually if not sooner.
When you are using the Ayre the secondary (output) winding of the Cary’s output transformer would have considerably larger voltages applied to it than when it is operated normally. And it would step up those voltages to much higher voltages than are normal on its primary side, and apply those much higher than normal voltages to the output tubes and related circuitry.
And when you are using the Cary, the Ayre could very conceivably also be damaged, as a result of the "reverse breakdown voltage" ratings of its unpowered output transistors being exceeded. (And of course having the Ayre powered up while the Cary is being used would cause other major problems).
This question has been discussed in a number of past threads, and you can find some suggestions in them of various speaker-level switching devices that are available. If you want to consider one of those, though, be sure it is rated to be suitable for use with amplifiers as powerful as the Ayre. And keep in mind that in most cases a tube amp having an output transformer should not be in a powered up state without a load being connected, especially if it is being provided with an input signal, and in many cases these switching devices may not apply a load to the unselected amplifier.
Regards,
-- Al