There is definitely some cyclical nature to older fashion. If you want to get very detailed you can find it constantly, but the large scale repeating happens less often. For example, you can see ribbons tying shoes in the early 18th and late 18th century, but between 1720 and 1785, buckles were popular and ties were for those who couldn't afford them. Small, but short term repeat. Long term we can look at the bum roll and farthingale popular in the later 16th century. The hoop that came into fashion circa 1700 began as a shape similar to a farthingale and eventually changed into an oval by the 1720s. The bum roll reappeared in the 1780s, still very close in shape to the original.

But, keep looking and the hoop appears again in the 1850s as the cage crinoline, eventually changing shape into the bustle. The bum roll also makes a come-back in as a bustle pad.

Other styles come back around because of sudden popularity of the time period, such as the trend for late 18th century styles during the Centennial. The Polonaise being a perfect example of this.

As for 1560 to 1590, the fashion changed a fair bit if you know what to look for. They would see the much subtler changes, such as the high neck giving way to a ruff or open collar and the breeches actually getting shorter, just as we can see the difference between slim fit and boot cut jeans today. Not surprisingly women's fashion is more evident: 1560 to 1590. They too go from high collar to ruff, their sleeves grow to enormous size, skirts grow wide at the top, and the waistline dips very low in front. Interestingly enough, the French hood style of the 1560s does seem to come back around in the 1590s, though it's not termed as such.

My favorite (and probably the most drastic) example of cyclical fashion is women's fashion after the French revolution. Regency/empire fashion for women was unique in that it was a complete abandonment of what had come before, and this was largely for political reasons. Think of the extravagance of Marie Antoinette, the corsets and panniers and wigs. As France attempted a Republic, fashion looked back to ancient Rome for inspiration and we see simple draped garments in pale colors with no corsets or hoops or panniers, hairstyles based on sculptures of goddesses with soft ringlets, and even a super short Greco-Roman-inspired haircut for some daring women.

This is the first example I see of a real, deliberate influence of fashion by a particular time period in the past.

Men's fashion at the time was also changing rapidly (with dandies, Beau Brummel, and the beginning of the modern suit), but not with the same historical influence.

This actually isn't a history question so much as a definitional one.

Definitional because you have to define under what criteria you can consider a fashion trend as "cyclical", because the looser the definition, the easier it is to define it as so because you can view it as a structural cycle (flowy to tight fitting back to flowy), but the more narrow the definition, the harder it is to define it as so because you can view it as a technological development (corsets vs. bras).

Because related to that definitional question, is the question of whether the events of time and history are cyclical, which I suspect are equally definitional rather than philosophical questions. The broader you define what constitutes cyclical, the easier it is to do so. The narrower you define cyclical, the harder, etc.

A more important question is, do you think it's cyclical now? From your usage of "always" you're making it seem like it is.