The other three connections
There are only four
Each side of a three phase transformer is connected either in delta or in wye, so there are four combinations in total. Delta delta, delta wye, wye wye and wye delta. Everything else you will read about, including open delta and zig zag, is a variation on these rather than a fifth type.
Delta delta
No neutral on either side. Used in industrial work where the load is three phase motors and nothing needs 120 volts. Its useful property is that if one winding fails the other two can carry reduced load in open delta, which is why it survives in plants that cannot afford to stop.
Wye wye
A neutral on both sides, which sounds convenient and causes trouble. Without a delta winding somewhere there is no path for triplen harmonics, so the neutral has to be solidly grounded and the system is sensitive to unbalanced load. It is used where both sides genuinely need a neutral, and usually with a buried delta tertiary winding to fix the harmonic problem.
Wye delta
Used for stepping up, and in some older industrial services. The delta secondary has no neutral, so anything needing 120 volts has to come from somewhere else.
Common questions
What are the possible connections for a three phase transformer
Four. Delta delta, delta wye, wye wye and wye delta. Each side is connected either as a closed triangle with no neutral, which is delta, or with one end of each winding joined at a common point, which is wye.
What is the most common connection for a three phase distribution transformer
Delta primary to grounded wye secondary. It provides the four wire secondary that buildings need and keeps triplen harmonics circulating in the delta rather than passing into the utility system.
What are the disadvantages of a wye wye transformer
With no delta winding there is no path for triplen harmonics, so the neutral must be solidly grounded and the transformer is sensitive to unbalanced loading. Many wye wye units carry a buried delta tertiary winding specifically to solve this.
How do I know if my transformer is delta or wye
Read the nameplate. It states the connection for each winding and usually shows a small diagram. A secondary written as 480Y/277 or 208Y/120 is wye. A secondary written as a single voltage with no second figure is delta.