What role does trustworthiness play in relationships?

Well, it's what you want it to be. It's not making your partner feel weird, bad, or different. Being a team. Get to know him/her and depending on the relationship do what you think you'd want, do not make them feel presured.

Answer

It is when you love and feel loved, you know that person will never hurt you intentionally, will not betray you, is faithful to you, is there for you when you need him or her, you are there for that person when he or she needs you, you have a special bond that helps create that trust and you cherish it and never take it for granted, ever.