Yes, you are correct. All weights should have been stripped off before balancing began. If you took them off at this point, the wheel would need to be completely re-balanced because the spoke weights figure into the total balance. If the spoke weights are located near the adhesive weights, then they were removed and simply reused.

However, if the weights are all on one side of the tire, then you may have a bigger issue. Oversized inner tubes will bunch-up at one side of the tire causing a heavy spot. When the tube is removed, it's easy to tell where the folds were. What you want is a tube that's correctly sized (or even undersized). I usually use 1 wheel weight, rarely use 2 wheel weights on a Airhead wheel (spoked or mag), and never 3. Did you pay for a new tube ? If so, what size ?

There should also be a 10mm diameter round paint mark on one of the sidewalls of the tire. That marks the "light spot" of the tire. On the installed tire... the light spot on the fitted tire should be aligned next to the valve stem of the tube (the inner tube's heavy spot). That match-up negates 1 or 2 wheel weights. Now that alignment is hard to do, but the location should certainly be within ~1 inch of the valve stem (to the left or right). Sometimes the painted dot is red, sometimes it's yellow, and it normally washes off after 6 months, but it should still be there on yours. ( https://www.tires-easy.com/blog/what-are-the-red-and-yellow-dots-on-my-tires/ )

And on top of all that, you want to make sure the bead of the tire popped out all the way around on BOTH sides. Failure to do this can lead to wobbles and shimmy at road speeds.

In short... tire installation is one of those things that really, REALLY matters to the rider. You should really learn to do this job yourself. Unfortunately, in a shop environment it's always left to the "low man on the totem pole"... the "new guy". And the "grunt" may or may not have gained this knowledge and expertise, which can jeopardize your safety.

Hope this helps.

@wobbly 

The adhesive weights (6 x .25 oz) on the front wheel are almost directly opposite the spoke weight on the other side. That suggests to me the adhesive weights are "offsetting) the spoke weight. There is no spoke weight on the rear, just 3 adhesive weights. I bought the tires through the shop. For spoked wheels, shouldn't they have asked me if it was ok to use adhesive weights? I expected them to use spoke weights...but I guess I expected too much? This is a big shop in S. Denver...tires were not cheap and neither was the mount and balance. I would have rather they just mount the tires and not balance if they didn't know how to do it correctly. Now I have to get the adhesive weights off, and pay someone else (that knows what they are doing) to re-balance the wheels. Do most shops have the spoke weights on hand?

Thoughts...

• I've worked at 3 or 4 shops and none of them had spoke weights on hand. We simply popped all the weights off at the start, and usually had some left over. They all went in a shop bin in case more weights were needed on the next bike.

• The predominant wheel these days in the mag, and only the adhesive type weights work on those, so it really doesn't pay for a shop to stock 2 types.

• If you paid a lot of money and you're not happy, why would you take it to a different shop? Take the wheel and receipt back and tell the shop manager you're not happy with the work. 

• You seem to be fairly careful about the care of your bike. And you should be because your safety is on the line, and riders these days need every bit of advantage they can get. At least to me, this comes under the heading 'if you want something done right you gotta do it yourself'. So again I would encourage you to learn to do this job yourself. Three good tire tools, a bucket, an air compressor and some good tire lube is all you need to change tires. Wheel weights and two 6203 radial (unsealed) 'plain jane' bearings is all it take to balance your wheel. 

I'm 71 and still changing my own tires. You can do this !

All the best.