How can I build customer relationships on Twitter while obeying the rules?

You need to be more specific with regards to "the rules"

From what I understand about developing customer relationships with twitter users, you need to constantly be on top with what people say about you. Check hashtags related to your company and communicate with your customers.

For example, a person wasn't happy with the experience they had with an airlines. A social media manager in charge of developing customer relationships would see his/her tweet and answer with an apology and offer him/her the chance to contact the company privately via Direct Messaging on Twitter or give that customer the company's phone number to better assist him/her with his issue.

Another example is a person who was happy with a company and that company would just retweet that person tweet and/or respond by thanking that customer for his/her service.

Relationships happen on one to one to basis.

In relationship you don't expose each other to the outside world.