I agree with SurferTim that the SPI wiring seems a bit strange.
I don't have one of these chips but your program does not make sense to me. What is on pin 9?
Your pulsing the SS pin low three time to change from I2C to SPI and then going on to write commands but do you maybe need to add another HIGH/LOW toggle to finish the SPI setup?
You then going on to read address 0x081A (Data Capture 0) that is a 16 bit register but you look to be discarding the result with writing the next two bytes and then reading the next 3 bytes that are 'Data Capture 1' and half of 'DSP core control'.
Also in the manual it states...
CONTROL REGISTERS SETUP
The following registers must be set as described in this section
to initialize the ADAU1701. These settings are the basic minimum
settings needed to operate the IC with an analog input/output of
48 kHz. More registers may need to be set, depending on the
application. See the RAMs and Registers section for additional
settings.
DSP Core Control Register (Address 2076)
Set Bits[4:2] (ADM, DAM, and CR) each to 1.
DAC Setup Register (Address 2087)
Set Bits[0:1] (DS[1:0]) to 01.