Can you post photos? I seriously doubt what you are seeing is what you think it is. It would also help to know where you live and what type rootstock the trees are on.

Will post pics when my husband gets home with his iphone.

I know it sounds bizarre, but I've been gardening for my whole life. We are fairly new to citrus, but these branches clearly change dramatically. Typical Valencia orange leaves and branches, then they turn thorned with the larger, sharper lime leaves growing at the ends of those branches.

We bought the tree at Home Depot - so I'll have to try researching beyond the tag that was on it. I guess it could be grafted onto Meyer lemon rootstock, but don't know yet.

Forgot to mention, we are in Ponte Vedra Beach, in N. Florida, east of Jacksonville and north of St. Augustine, on the coast -- 1/2 mi off ocean. The fruit trees are west of the house in a protected area backed by pines and forest.

1. Cross pollination cannot change an existing tree in any way. It is only seedlings grown from a cross-pollinated fruit that could be different from the original variety.
2. Except in extremely rare cases (called sports), branches do not change variety - and never from orange to lime.
So, as Johnmerr suggests, I think you are mistaken. New, young growth on citrus trees can look different from more mature growth. Thorns can also occur on vigorous new shoots.
The only other possibility is that you are looking at root-stock branches that originate below the graft point. These will indeed be different from the orange variety, but can't suddenly change along their length.

Could be water sprouts?

It sounds like you're having rootstock sprouting up from below the graft, as Citrange suspects. Nearly all citrus are grafted - the cultivar you've purchased (in your case, a Valencia orange) grafted to some other citrus variety as the rootstock (which provides more resistance to soil conditions, pathogens, and regulates growth). When you post photos, can you post photos of where these different branches are emanating from at the trunk? We'll try to tell you if they're coming from below your graft line. Also, the leaves on these different branches - are they in groups of 3? If so, then for sure this is rootstock, as most of our citrus are on some sort of trifoliate rootstock. Also, what makes you think they are lime? Do the leaves smell like lime when you crush them??

Patty S.