President orders National Guard to Southern Border.
NM Governor orders National Guard to stand down.
Whose orders do the Guardsmen obey?
That would depend if they’ve been federalized, i.e. mobilized into the Army, or not. Until they’re federalized, they’re under the sole control of their state.
For a historical example, look at George Wallace’s attempt to use the Alabama NG to stop black students from attending the University of Alabama in 1963. President Kennedy federalized them, and then ordered the same National Guardsmen to enforce the Supreme Court ruling.
That said, I suspect Trump is just bloviating, and the Governor of New Mexico is telling the New Mexican NG that until they’re formally federalized, they answer to him, NOT the President.
The National Guard deployed to the border in the current mission are basically all under state control, but with the Federal government paying the bills. California has made a similar decision that it will only support certain parts of the border mission, so New Mexico appears to be in an identical posture as California.
Note that the National Guard when acting in a state role has various law enforcement powers that they lose if called up in Federal status. Really, those law enforcement powers don’t seem especially relevant for the current mission, but it is something to be aware of.
Assuming we’re talking about the New Mexico case, the first part of your post doesn’t fit what happened. Trump never ordered the NM National Guard to the border. The National Guard troops that have been performing border security support mission have been on Title 32 orders.
Title 32 orders are a bit of a hybrid between the state and federal statuses that the National Guard can be utiilized under. On Title 32 DOD makes money available to the states for specific missions they would like the states to perform. The troops are still under state control on Title 32 and the Governor is still their commander in chief. The Governor can refuse the mission or end the mission anytime they want. The President can order DOD to withdraw funding and end the mission. The title 32 hybrid allows National Guard troops to keep their exception to Posse Commitatus. They aren’t federal troops under Title 32 so they aren’t constrained domestically like federal troops are.
The New Mexico National Guard activated troops on the order of the Governor, at the request of Trump. The federal government paid for the mission. The Governor just changed the orders. The National Guard troops go home unless something changes.
That something would be a lawful mobilization order that federalizes elements of the New Mexico National Guard. At that point they are federal troops and follow the lawful orders passed through the chain from the National Command Authority. At that point, as federal troops, they also can’t do anything that any other active duty troops can do. Posse Commitatus would apply.