What’s with the westernmost tip of Texas (north of El Paso), and its border with New Mexico? It looks like the kind of shape that a river might create. Was this the original course of the Rio Grande, and at some point it shifted?
(Sorry, I’m having trouble linking to the map.)
Well, here’s a link to the map, at least. I don’t have a definitive answer to your actual question, unfortunately — but I’ll note that the river looks unnaturally straight in the region I’ve noted, so I would guess the shift was man-made.
From here Bing Maps - Directions, trip planning, traffic cameras & more
It’s pretty clear there is a road named “Levee Rd” which runs right alongside the river from just south of Las Cruces to nearly the border. Good bet that road has a levee next to it.
A little more zoom in, playing with satellite & bird’s eye mode, and it’s pretty obvious that the current river course is not the natural one.
Your guess is basically right - the Country Club Dispute:
There was also the “Chamizal Dispute”
That’s the border between Texas and Mexico, not Texas and New Mexico. 
panache45 said “westernmost tip of Texas”, which hits both Chihuahua and NM. But, you’re right. I wasn’t paying attention. I worked graveyard shift last night, and its way past my bedtime.
What originally caught my attention was the fact that the streets and property lines in that area totally disregard the boundary, which actually goes through some buildings. How do they work that out regarding taxes, voting and the census?
Interestingly, you can still see clear evidence of the former course of the river on the aerial photos. Look here, for instance - you can see the deposits left by the river’s meanders. Same thing a bit further north here.
On that second map, Rosebud Lane, a small cul-de-sac with only about nine properties, appears to start in TX and project out into NM. Seems pretty weird for such a small street to span two states.
I was going to say that the borders shown on Google Maps are only approximations, and indeed that is the case for much of the world, but it appears that in the USA they are actually very accurate - compare the topo map for the same area. The border is made up of a series of straight lines linking monuments - which are unfortunately too small to show up on the aerial photos.
Take a look at some maps of Anthony.
On paper, there are two towns: [Anthony, TX] and [Anthony, NM] but it’s actually one community which straddles the border. The NM tax authorities are always grumbling about people living on the NM side, but registering their vehicles on the TX side. It would not surprise me if there are a lot of streets like that.
At the risk of ethnic stereotyping, I would speculate that the houses on those streets are owned by Hispanic families that have been there since colonial times. The streets, and probably many of the houses, pre-date the current boundary lines.
The house probably has two mailboxes, one on the NM side and one on the TX side. Uncle Joe is the owner of record on one side, and Uncle John is the owner of record on the other side. The titles probably show exactly how many acres are in which jurisdiction.
The nephews who live there list their official address in whichever state is most convenient. If you work in one state, you register in that state, so you only have to file one state tax return.
NM has worse drunk-driving statistics than TX, so a NM address means higher auto insurance premiums. So occasionally, people will register their vehicles in TX. If it turns out that your garage is on the NM side, the NM Tax&Revenue people get very irate.
Straddling properties are not too uncommon around most state boundaries, particularly those which are an arbitrary line rather than following some natural feature, and nobody’s chosen to build a road right along the border.
For instance, I just arbitrarily pulled up Maryland, and examined the PA/MD border. Here’s a section north of Hagerstown, with the state border bisecting buildings:
The business or residence involved is usually considered to be in one of the states, in spite of the property involved spilling over into the other.
Oh, and if you want to quibble about the accuracy of border placement in Google maps, remember that that was a choice made pretty much at random. If you search around you will find better examples where state lines clearly bisect properties even if the line isn’t very accurately represented. Even on that one, if you scan east a bit, you will notice the border running right through the middle of a lot of farmers’ fields, and right smack through a few sets of farm buildings.
Without addressing any of the rest of the post, let’s lay this puppy to rest now.
The boundary will have been established at a time when the Rio Grande flowed along the route the boundary marks. There has been a subsequent shift in the flow of the river (from the looks of things, mostly due to the installation of a leveed canal for the river). The old boundary will have statyed the same even with the move of the river; that’s the way such boundaries are handled when rivers change course due to an individual event (see the weird Mississippi River boundaries for examples.
Thus, when the boundary was set, there were no structures there, since they would have been built in the middle of the river. Any structure straddling the line was built after the boundary was established.
Actually, I’d guess that most of the difference was from a natural shift between 1850 and 1912, as per the “country club” dispute - nobody was sure where the river was in 1850, and they didn’t have aerial photos to help determine it. They just knew that it had moved a large amount between then and 1912. When they did the levee, they probably more or less just straightened the river out, leaving it roughly where it was. It would help to know when the levee was constructed.
DSYoungEsq, the river is not the only boundary line. Nearly half of present-day New Mexico used to be part of Texas. County lines have shifted drastically in the last 150 years. Municipal boundaries are constantly shifting. Sometimes one group will split off from a larger group. Sometimes a larger group will annex their neighbors. Either way, there will likely be a few property owners who end up with boundary lines crossing their backyards.
The Hispanics have a joke, “We didn’t cross the border. The border crossed us.”
The Rio Grande is the boundary line in question. Unless you are referring to the 32nd parallel, which establishes the southern boundary of New Mexico with Texas, which is north of there and runs east to west, which isn’t what was under discussion at all.
What IS under discussion by the original poster panache45 is the boundary between Texas and the New Mexico portion of the Gadsen Purchase, which extended the border of the United States south of the 32nd parallel a few miles. This resulted in a border between Texas and the New Mexico Territory that ran for something like 12 miles along the Rio Grande. Which is why I said what I said.
And, while Texas claimed a large portion of what is now New Mexico, that claim was given up in 1850, IIRC (for the sum of $10,000,000). I think you will find very little in the way of structures in the area dating to before 1850, outside of Santa Fe and Albuquerque.