So, is this something new that is happening? Have you noticed it before or has something changed that this feeling is there. There is something called 'hard flaccid syndrome'. https://www.urologynews.uk.com/features/synopsis/post/hard-flaccid-syndrome It's a little off but does have some symptoms you describe. Do you have pain upon ejaculation?
I'm not sure this is a rigidity problem, but some men do have certain things that can be just the way you are or could be a sign of a problem. Only seeing a specialist would probably tell you that.