I am an anxious attacher with an avoidant attacher husband.
We have made quite good progress recently but we are BOTH having to work at it.
I found a really good article that I sent him a while back that describes him to a tee. I'll see if I can find it. It was very helpful in understanding him. He found it helpful too in understanding himself and recognising some of his behaviours (avoidant attachers can be a bit oblivious as they deal with things by ignoring, packaging them away).
We spent a good chunk of our last holiday talking about it and how he feels and how I feel. And trying to think about how we could move forward.
What's interesting is both attachment styles come from a place of insecurity. When I feel insecure I feel anxious, lost, without anchor, sometimes panicky. I feel like the rug has been pulled from under me and I am emotionally flailing. It makes me want reassurance and connection to make me feel ok again (although I do sometimes push him away too).
When he feels insecure he thinks "well the only person I can rely on is myself" and he shuts down emotionally and pushes it away. His superpower though is that he can then just get on with his day as if nothing has happened! But is super distant and not connected to me at all which feels pointless and lonely from my perspective.
It was quite difficult to get him to understand that there was any point to connecting properly and being vulnerable with someone as he'd never really done it so hadn't experienced it. He was quite capable at getting on with things and his life without this "connection" and he wasn't sure what the point was. I managed to persuade him that he was missing out and after a LOT of talking we've made some quite good in roads. We do have mis -steps (when things can flare up super quickly!) where we revert back to old communication styles but we seem to be getting quite good as getting things back on track when this happens.
A lot of what we are doing is trying to listen to each other without reacting. If the other person says something perceived as hurtful, the other person tries not to have that knee jerk reaction but instead check in with what they meant. Calling each other out if we revert to old communication styles. Reassuring each other that we love them and making sure that he knows he can rely on me. When my husband can see I am feeling unanchored he's become quite good at holding me and reassuring me, whereas before he'd shut down or withdraw and make me feel so much worse. Interestingly when we talked about this it was because when I was struggling it makes him feel guilty and as that felt bad he'd shut down.
The other thing we've said is to be completely honest with each other about how we feel. DH has a tendency to not say how he feels until it boils up and it'd all come out at once, or leach out as resentment. I would not say how I felt as I worry about him detaching. So we've stopped that as that's obviously really unhelpful as neither of us knew what was really going on with the other which led to a lot of misunderstandings and arguments.
I've also worked a lot of self-soothing.
Anyway not a great pairing as others have said but also remarkably similar underneath all the defence mechanisms we've set up for ourselves. I think it is possible to work on it. But it needs to come from both of you. And it's probably going to be a constant work in practice. Personally I don't think it's possible just to do it from your side, as an anxious attacher myself I think that would be too hard.
I'll post some links below to some resources I found really helpful...