I know cacao, rubber and turquoise were traded, but what else? And was trade regular among these peoples? I know the Mesoamerican ball game was brought to the Southwest, so I assume the cultural influence from Mesoamerica would have been extensive.
The primary artifacts of trade between the Southwest and Mesoamerica were ritual items of high value. This would have included copper bells and Pyrite mirrors coming up from Mexico. There isn't a tremendous amount of analysis on non-prestige trade regional trade, although it definitely occurred to an unknown degree. Such trade would have been short-range, neighbor-to-neighbor rather than the rather longer trades prestige items required.
The regularity of trade varied with the time and place, but at certain points (especially at the height of Chaco in the 11th and 12th centuries) there would have been a comparative flurry of trade with certain wealthy cities like Tula. There have even been limited suggestions of trade in utilitarian items like obsidian flakes, but all of these have been "identified" without proper testing to my knowledge.
Cultural influences are rather more complex and entire books have been written on the subject. There was certainly a good deal of back-and-forth between the regions, but the influence was quite nuanced. The word commonly used in the literature is "semiperiphery". That is to say, the southwest was in contact with mesoamerica and ideas flowed between them, but was not dominated or wholly in step with it.