Can someone with Alexithymia be an extrovert?

So I've recently been diagnosed as alexithymia-tending, and I feel like a large part of the underlying confusion over my life has disappeared! However, I've realized that I've had alexithymia as far back as I remember, and my dad probably has Aspergers. I never realized I felt differently from other people because there were no flags; I thought I was just a nerd, and more importantly, I never had any major problems socializing, plus I love to make other people laugh or be happy.

I read somewhere that alexithmyia also makes it hard for us to recognize emotions in others. Am I just not as good as socializing as I believe? I thought I was pretty likable because I'm good at directing and mirroring the emotional flow of things, and it would be pretty awkward if I had only imagined it...

Though I do have problems getting close to people I don't really care to get close to (since I don't seek out company unless I like that person), I've never had problems maintaining a long-distance relationship or making friends with people who I immediately like. It's easy for me to do people-facing work, too. Does this mean I might not have alexithymia?