Core Ground Bushings on Top of Transformer?

Normally two bushings are brought out from core structure. One from core and other from core clamp. These will be shorted and earthed to tank.

I dont think X1-X3 shorting is additionally required.

External core ground connections make it much easier to meggar the core to ground insulation. Does the transformer have an LTC? If so, the second bushing is for the LTC autoreactor core. They should both be connected to ground.

The second bushing may have the ground removed on purpose. I agree with stevenal that in normal operation both the core and the reactor core should be grounded. Check the resistance to ground with a fluke on the ungrounded bushing. There may have been a second (unintentional) ground at one time and the core was grounded (single point) elsewhere. Do you read ground or a few hundred ohms? Does the owner have any history?

With the exception of the LV Excitation Test, I can't think of any other standard tests (TTR, Megger, Power Factor) that require X0 to be grounded during the test.

1. I also agree with the general consensus. "Megger" that one floating bushing against ground to see what it's value is.
Any Core ground or Frame (clamp) ground that is brought out should go to earth.
If it was left open due to an inadvertent ground then that bushing should still go to ground thru a large resistor because if the inadvertent ground clears the Core or Frame will be floating.
I would also take a close look at the DGA of that transformer to see if an issue exists.

2. The jumper from X3 back to X1 just ensures you have all three phases properly shorted together.
Think of it as creating a "Delta" connection out of the Wye terminals vs. a "Open Delta" connection with just shorting X1-X2-X3.

3. I agree with DTR2011. LV excitation on a Dyn is the only test I can think of where the X0 should be grounded. Be sure to state "X0 Grounded" in the notes section of that test.
The LV lead will be on the H0X0 of an autotransformer during LV excitation.
When testing winding resistance, CTs and TTR the X0 will often be in the test circuit or floating when not.

Thanks for the responses. I did find this document recently which states that one terminal of each winding in a resistance test must be grounded:

Section 7.1

Since a wye winding with neutral will have all measurements taken between the phase terminals of each winding and neutral, I assumed the neutral ground should stay put to prevent it from floating to a high voltage.