I'm sorry to hear about your mother. My mother died 5 years ago. She struggled with illness for a long time but I, too, didn't think she could actually die; not in her 60s. I miss her, I grieve her loss. There's still a hole in my life that's shaped like her. The thing that's gotten easier in the last few years is that the edges of that hole aren't as sharp. My memories of her and my sense of her have become more golden and light. Thinking of her is often a joy instead of just a gut-wrenching pain.
One thing that I learned recently about grief which really helped me is this: You don't have to stay in pain to prove that your love mattered. In time you'll be surprised to find that some healing has happened, the loss isn't always as brutal. It feels weird. It feels like a betrayal, but it's not. It's just life and the passage of time, it doesn't mean that your mother wasn't wonderful and you didn't love her dearly. I know a month isn't long and it may be quite awhile before any of this sounds conceivable, but these have been my experiences and what has helped me. I am sorry for your loss and I hope that you and your twin are able to receive the love and support that will most help you now.