As far as I can tell, when the TRS3238E goes into powerdown mode, it fails to propagate the start bit from RIN* to ROUT*:
Tx Rx ROUT* (not including the stop bit)
---- ---- ----------------------------------
0x00 0x80 136 us
0x01 0xc0 122
0x02 0x81 136
The ROUT*, at 57600 bps, should be 156 us, not including the stop bit.
In the "0x01" case, since we miss the start bit, and the LSB is "1" (RS-232 is LSB first), the whole thing is off by two bits.
Jim