If transitioning into academic position, should I buy my PhD graduation cap and gown? My graduation ceremony is coming up in a few months,
and I have the option of buying or renting a graduation cap and gown.

I will be starting an assistant professor job this year, so I suspect (but am not sure) that I will need to wear the dress at graduation ceremonies at my new institution.

If I were to buy the graduation dress, would I ever need to wear it again?

Location: My new job will be in an Asian university.

What I did I asked a colleague at the institution where I will be working,
and he told me that indeed, I could rent a gown for free
if I were to attend a graduation ceremony.
As noted in the comment, the location might change our opinion of the problem, however I will make notes from my experience.

When I graduated with my first degree I hired a gown as I could not see the value of owning one and knew I would study for higher degrees, and perhaps I could buy one later in my career. I received my second degree in-absentia because I knew that I would graduate again with a third degree in due course.

Now I attend ceremonies on a regular basis my institution can provide my gown for any formalities. Someone really parsimonious might say that you never need to buy a gown. 

However, I have great regret not getting the earlier gowns. Time has changed things in a way I could not predict when I was making the decisions. The gown from my first university was designed by the hippest fashion designer of the 1960s. It was real 1960s cool in colour, shape and cut. No black gowns and mortarboards for them. As the institution matured they felt hip fashion icons of the 1960s were no longer cool and switched to plain black gowns and mortarboards. Now, in present times, it is impossible to source the original gowns. They are collectors items that rarely come on eBay and go for huge prices. All the original graduates, like me, now realise what a fantastic item they missed and want to get them. (Because, today, they look quite fantastic again).


The Computer Science Class of 1976 - Note the curved Hats

The gowns from my second university, which some of my colleagues own, have changed in quality over the ages. When I graduated they were made by fine tailors from excellent cloth with fine silks and quality trimmings. Today they are polyester and so forth and mass produced identically to all the other university gowns.

My rented gown just does not impress half as much as the originals that others wear at these fine ceremonies.

Only you can know how fine are the gowns that you could buy. No one can know what the future brings.