First, you will listen closely to what your partner has to say.
According to my fortune, my future partner is a prince in some country. I wonder if it's true.
I don’t want a stupid talking companion.
Your mutual hand is someone else's hands (or just someone else, if you remembered that 手 can also refer to a person as well). Whose hands are these? These are your partner's hands. That's why everything is mutual.
The readings are the kun'yomi readings. For 手, that's no problem for you. For 相, that's more of an issue since you learned the on'yomi reading for it. Think of it this way:
Your partner wouldn't be your partner without you. Your partner is: I (あい) and you. Remember that as you gaze into your partner's eyes. "I and you."