If transitioning into academic position, should I buy my PhD graduation cap and gown?

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.

Don't bother. It's over 10 years since my ceremony, I've been in academia all that time, and I've never seen the need. I've also seen very few faculty with their own robes. For graduation most institutions I know of will organize robe hire (and pay for it), so there's not even that incentive.

Different institutions have different policies on robes for faculty members at graduations, although most will arrange for robes for faculty who do not have their own. However, if you take advantage of this option, you might not get much choice in your regalia - so if the institution that you are graduating from has its own particular design for regalia that you love, it might be worth getting your own. I have heard that there are institutions who will require that all faculty members wear the same design of regalia for graduation ceremonies, in which case spending a lot of money on a non-traditional set - which you then cannot wear at commencements - would be less than optimal.

Since you have an assistant professor job lined up, you could just ask someone at your new institution what the policies are there.