Ah.

Pesky fine print, in the datasheet for the PCA6416A:

There are two supply voltages for PCA6416A: V_DD(I²C-bus) and V_DD(P). V_DD(I²C-bus) provides the supply voltage for the interface at the master side (for example, a microcontroller) and the V_DD(P) provides the supply for core circuits and Port P. The bidirectional voltage level translation in the PCA6416A is provided through V_DD(I²C-bus).  V_DD(I²C-bus) should be connected to the V_DD of the external SCL/SDA lines. This indicates the V_DD level of the I²C-bus to the PCA6416A. The voltage level on Port P of the PCA6416A is determined by the V_DD(P).

Hmph.  This does explain why my initialization wasn't working; I was trying to configure the chip before turning on V_DD(P), and not getting the ACK.

But it really would be nice to be able to initialize the chip completely before turning on I/O power and thereby enabling dangerous things.  I guess I'll have to make sure that the power-on-reset state doesn't lead to any chaos.

Hmmm, maybe I should (for some cases, not the Gadget currently on the bench) provide a 1.65V keep-alive supply on V_DD(P) for those times when the I/O supply is powered down...?