I have a great relationship with my 16yo daughter - she would definitely say the same - and a pretty good one with my own mum.
I think it helps that we are very similar in terms of how we think about stuff - she often says that she inherited my brain as our thought patterns are very alike. I guess that's luck really, more than anything.
I think she knows that I'm always on her side and that I'll always be her biggest cheerleader - that probably helps! I truly think she is amazing and she absolutely knows it! 
She has very good self esteem and I like to think that I've fed into that by constantly telling her all of the things that I like about her.
I have always tried to listen and respect her point of view - none of this "I'm the adult and I said so" nonsense. I treated her with respect and dignity, even as a small child. If I wouldn't let her do something, I always explained why, and I was open to hearing her opinions and being persuaded sometimes if she could make a good case. I allowed her to make as many decisions about her own stuff as possible, while ensuring that she would be safe. I think this has helped to minimise the risk of teenage rebellion because she sees me as fundamentally reasonable.
We talk a lot. Endlessly and about everything. I know loads about her friends and her ideas and her aspirations - I'm genuinely interested and we both really enjoy these conversations. She knows quite a lot about my life as well, including some quite personal stuff about the things I dislike about myself etc. I have always been very open and honest with her about things, in an age-appropriate way. My mum kept lots of secrets to try and protect me and I always hated that she wouldn't just tell me stuff.
I don't really nag her about stuff, I just try to support her in being the best version of herself that she can be... but on her terms, not mine. So I'll ensure that she is awake if she has decided that she wants to get up early but I'll leave her to sleep if not.
We have talked a lot about values over the years, and we are on the same page with these, but I try not to impose my ideas too much, and I respect her decisions even when I don't like them... when she got her nose pierced, for instance. She knew that I didn't want her to get it done but she also knew that I believe in her right to make decisions about her own body. I went with her to the salon and actually paid for her first stud. I think it's really important to accept that our children are separate people in their own right and not extensions of ourselves.
We laugh a lot! I try to take an interest in whatever she is into. When she was little, I had to fake it a bit, but it's easy now and I find it interesting watching the things that she wants to watch, listening to the music that she likes etc. I have always really enjoyed spending time with her, which helps!
I have always made it clear that adults/parents are not always right, and I have made a point of apologising when I have done something wrong. DD says that this was really important to our relationship, because she was never under the impression that we were perfect, whereas some of her friends put their parents on pedestals when they were very young and then felt angry when they realised that their parents were just flawed human beings like everyone else. DD never had the illusion that I was anything but flawed so it wasn't a disappointment!
If dd does something wrong, I'll talk to her honestly about it - about the consequences of what she has done and about how I feel etc - but I have never imposed sanctions or punishments of any sort and haven't ever needed them.
In a nutshell, I think it boils down to love, honesty and respect on both sides. And plenty of fun!