Sharing Clock and Data for 74HC595 and CD4021B Shift Registers?
If I may answer my own question.  It should work.  Here's how with arduino over SPI.

void setup() {
    //Setup SPI
    SPI.begin();
    //Setup CD4021
    pinMode(CD4021_PSC_PIN9, OUTPUT);
    digitalWrite(CD4021_PSC_PIN9,HIGH);
    //Setup 74HC595
    pinMode(HC595_LATCH_PIN12, OUTPUT);
    digitalWrite(HC595_LATCH_PIN12,HIGH);
}

void loop(){
    //Shift out to 74HC595
    digitalWrite(HC595_LATCH_PIN12,LOW);
    SPI.transfer(BYTE_DATAOUT);
    digitalWrite(HC595_LATCH_PIN12,HIGH);

    //Shift in from CD4021B
    digitalWrite(CD4021_PSC_PIN9,LOW);
    BYTE_DATAIN = SPI.transfer(0x00);
    digitalWrite(CD4021_PSC_PIN9,HIGH);
}


Learned from here, from someone who has done this exact implementation before.