You and he have built a life together, whereas his relationship with her is very new and who knows if it'll work out long-term yet? Therefore, for him to want everything to be equal between you and her is not reasonable. It sounds like he's in the throes of NRE, or " new relationship energy"... basically he's letting himself get caught up in all these strong emotions and hormones and not acting rationally. He needs to recognize this, make amends to you for the lying, and take things a lot slower.
I would encourage you both to check out the How-To Guides section of this excellent website to begin to get some basic principles in place as you move from swinging to rule-breaking to, hopefully, actual ethical poly: http://www.morethantwo.com/
Best of luck!
Me, 30ish bi female, been doing solo poly for roughly 5 years. Gia, Clay, and Pike, my partners. Davis, ex/friend/"it's complicated." Eric, Gia's husband. Bee, Gia and Eric's toddler.