If I understand you correctly, you are describing a "wild-leg" delta system. They are nothing but trouble.
You start with a three-phase delta connection, with the phase-to-phase voltage of 240V. So far, so good.
Now, you center tap ONE of the transformer secondary windings and ground it. This gives you 120V from each adjacent corner of the delta to the center tap (or ground). So you really have a three-phase 240V system, plus two 120V hot legs tacked on.
So, the ability of the transformer to handle 120V loads is quite limited, because you are working with only 1/3 of the total transformer.
This is often used for irrigation systems and pump stations where the predominant load consists of the three-phase motors (230V) and 120V load is limited to a few control devices, lighting, etc. If you have mostly 120V loads, you don't want to use this.
If you need three-phase power, and have a choice, take service at 480V and buy a little "uni-center" to step down to 120V if necessary. Then you *could* have, 480V (1-phase and three-phase), 277V single-phase, 208V (1-phase, and three-phase) and 120V single phase.
Utilities sometimes encourage these wild-leg systems because they can use three pole-mounted single-phase transformers that they probably have on hand in their yard. 480v transformers are usually not on hand.
NEC has some special requirements for the wild-leg systems.
If you need more info, let us know.
dpc