ZAB Controller described a confused hand off/coordination event between ZAB and MZT Center; suggesting that improved procedures must be develop between the two facilities to accommodate aircraft transitioning the boarder.

Narrative

I was working Sector 46. Not a whole lot going on; although there was a lot of weather along the border from OLS; East and West. I heard MZT call Sector 91 for coordination. Thinking there was arrival traffic coming from there and knowing they were deviating; I expanded out to see who it was. The route from MZT's airspace is always VYLLA.J92.TUS; then points North. Because of this; all flight plans over VYLLA are routed through Sector 46 for coordination through the A-Side into and out of Mexico. I happened to see a limited target (Mode C intruder) West of the DUG VOR at FL360 and said something to Sector 90. Sector's 90;80; and 47 were combined Sector 90 and were extremely busy with weather deviations and hadn't seen the limited data block. MZT was in the process of; presumably; coordinating the deviating traffic; but with the wrong sector. Eventually we figured out it was Aircraft X who called them up after MZT coordinated with 91 to have them shipped to 90. Recommendation; the problem with deviating traffic over VYLLA is we don't do hand offs. It's all Non-RADAR. The flight plan goes through Sector 46; although 91 or 65 will have the appropriate URET information on the flight plan. If the aircraft is deviating East of VYLLA; it puts them in 90 or 80's airspace depending on the stratum. Without any coordination; they can't protect for the aircraft because they have no knowledge of the aircraft. I think the simplest thing to do is to do hand offs between us and MZT. Also; both facilities need highlighted information on altitude stratum's and sectorization. MZT isn't aware of any sectors other than 46; 91; and 65. Most of our controllers aren't aware there are two sectors at MZT along the border. Non-RADAR separation and protection only works along the route. There are obvious issues with deviating traffic; especially in a sector who doesn't know the aircraft is coming.