Different Tertiary institutes have different requirements for which regalia to wear for conjoint or double degrees.
University of Auckland - Conjoint Degree Set
What to Wear
Bachelor Gown, Degree Hood, Trencher
Students graduating with a conjoint degree are able to graduate at a single ceremony of your choice and wear the one hood for the degree appropriate to that ceremony. e.g. Graduating with BCom/BSc, attending BSc ceremony, must wear BSc hood only.
You may also hire or purchase your second degree hood which can be worn at all other times, except for at your graduation ceremony
AUT - Double Degree Set
What to Wear
Bachelor Gown, Conjoint Degree Hood, Trencher
Note: One burgundy lined conjoint hood will be included in your order.
Outside of the graduation ceremony you are permitted to wear your actual degree hoods appropriate to your degree for photographs and the procession.