It's all brand new, barely used at all. No smell with the heat, just the AC, yet they use the same ducts and vents. That has me stumped more than anything.

Could be just the normal burning off of oily residue that new HVAC is packed. Source of smell should be easy to locate if it's as bad as you seem to indicate. I would be surprised after several weeks that smell has not disappeared. You never did tell me the specifics of your new system that I requested. Blower motor could be a suspect.

IMO

There are a lot of questions unanswered. First in my mind is what is the nature of the offensive smell? Is it chemical/oil/smoke, or is it musty/moldy? If the latter, does it smell like the basement?

Does your system really bring in outdoor air? I'd venture to say that yours does not. Some bring is a small proportion of outside air and recirculate most. If you have new ducts that leak so much that they are sucking air in from the basement or elsewhere, the installer is at fault.

Since you only have odor in cooling mode, I suspect that you have microbial growth in, around or under the evaporator coil. In the summer, however, in most places, more things are moist and support microbe growth including basements and crawlspaces. That means that the odor could be coming from other places. As tiger dunes suggested, someone needs to look at the evaporator coil and see if something is growing there if you have that kind of smell. (Look up "dirty socks syndrome"). Post model numbers of your equipment and, if you can, some pictures of your furnace, ac unit and ducts.

Disagree with above post...brand new system...how could this smell come aboard in such a short time...as far as cooling mode, system is so new that heating mode has never been operational...

IMO

No, no. I'll try to be clearer. The Trane AC system was put in in May of 2104. I also, at that time, had a new furnace, also a Trane, put in. The ductwork did not change and was the same ductwork that connected to the furnace before this installation of AC and the new furnace. There was no AC prior to my having it installed in May 2014.

The first time I turned the system on after installation, there was a very loud explosion. Long story short, it was the coil on the AC and it had split at the seam. This caused the AC to not cool the house last summer. HVAC guy got the new part last summer and fixed that and then the AC system cooling did work.

But it was also stinking up the house. Don't know how to describe the smell. HVAC guy send some guys out to look at the ductwork for leaks and they said it all looked good, didn't find any leaks. We did seal up a very old vent on the duct (??not sure of the word) that rises above the furnace as we thought that was the source). So everything is supposedly very well sealed, no leaks in the ductwork, and the weirdest and most telling clue is that there is NO smell when the heat is on, which I obviously had on all winter, but only when the AC is on, yet they both use the same duct and vent delivery system.

I also have a very good dehumidifier in the basement, which runs 24/7 and drains automatically. The basement is very dry.

If air was somehow being pulled in from the basement, would it also smell when the heat was on? Why would an evaporator coil have microbial growth on it when it's so new? Remember, this smell has gone on from the VERY start, upon installation.

Bats crapping in the attic. They arn't up there during winter because they migrate South. Smells somewhere between kangaroo rat farts and wino morning breath. Better leave them alone,they are protected under five different federal statutes. The coil spring explosion is a mystery,I just hope the replacment doesn't blow up to. I believe more guests would be scared off by coil spring explosions than from bat guano odor. Keeping cats in the attic will take care of bats but then cats have to go also and that might smell worse than before.

klem, how do you know what wino morning breath smells like?

Curious.

What's the typical outdoor temperature and humidity? Where is the airhandler installed, and what is the T&H in that area? What does the smell smell like?

I had a warm, humid, swampy smell when the AC first started cooling after being off for a while. The AH is installed in the attic, where T&H could get up to 140° and 90%. We eventually decided the attic air was condensing on/in the cold AH case, forming a puddle inside the AH. The AH has a drain, but it's a poor design/installation, allowing up to 1/4" of water to collect in the bottom of the AH before reaching the drain. As long as the fan is running, the air circulates, but when it stops, the puddle evaporates, feeds any mold or algae around or in the ductwork, and forms a glob of stinky air that blows into the home when the the fan starts up again. There wasn't much evidence to support this (inside of AH smelled fine), but we super-insulated the AH, vented the attic a little better (now we see 100° and 90%) and the smell mostly went away.

Update: Quite by accident I found the term "dirty sock syndrome" which is almost certainly what I've got (or had). Mold accumulates on the evaporator coil, and smells on startup. Most-recommended fix is to install a UV air purifier in the airhandler. @d2305: no standing water these days, evidence that there has been in the past.

Look at the pan under the A-coil for standing water.