Answers to questions like What should the rarity rating be for this homebrew Healing Brick? are usually based on comparison of effects of an item. On the other hand, there are questions like

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  The power level of the Sword of Sharpness doesn't justify its very rare rating - am I missing something?,


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  Handy Haversack vs. Bag of Holding and


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  Why is this uncommon magic item better than this rare magic item?

that make me doubt if rarity is really tied to power^*.

Is there any rule or guideline that says more powerful items should have a higher rarity (or that rare items are more powerful than less rare items)? For example, if I have a legendary item can I say for sure that it is more powerful than items in the lower rarity tiers? Or can rarity also be indicative of other factors besides power?

I'm looking for general rule or guideline, or lack of it.

* If definition of item's power is needed, use the same definition that is applicable to the word on DMG p135.

There is a rule but it is very loose

The DMG says (p135) (emphasis mine):

Rarity provides a rough measure of an item's power relative to other magic items.

The emphasis is on rough here as the designers are not applying this as consistently as you would expect. Examples of this are in András' answer and nitsua60's answer and the cases as mentioned in the question

I normally play a lot in the Ebberon setting where the rarity of magic items is a bit different anyway. Therefor I spitball rarity and price a bit by looking at similar items but heavily modify it by distance to large manufacturing centers and such.

Rarity and usefulness/power are very weakly correlated, in my experience.

This has been discussed elsewhere (reddit, GitP, ENWorld), primarily when 5e was first released. I'll point you to my favorite resource, the Sane Magical Prices Index by GitP user Saidoro: I've used it for years in order to gate items in campaigns and have been very happy.

First Saidoro establishes that item rarities are obviously bunk in plenty of cases: compare (for yourself) the broom of flying/winged boots to the wings of flying or boots of levitation. We see there functionally-equivalent or even weaker items "rated" two tiers above comparands.

After dividing the items into comparable classes (consumables, combat items, utility items) Saidoro and other posters spent months discussing/debating the utility/power of each and set a scale, in gp, for almost every magical item in the DMG. Follow the threads both on GitP and on reddit/ENworld (linked in Saidoro's .pdf) if you'd like to see more of the reasoning that goes into each ranking/valuation.

The long and the short of it is that while one might argue with some of the valuations, there's no question that this list, compiled by many actual players and much more finely-graduated than the rarity tiers, is a better guide to power than is rarity.

So I threw all the prices and rarities into my statistics software and ran a regression with rarity as the explanatory variable: the R² value is ~0.0236. That tells us roughly 2% of the power/utility of each item--as judged by users and tabulated by Saidoro--is attributable to the item's rarity.

Not really

Beside your examples, a Flame Tongue (rare), is much better than a Frost Brand (very rare)¹, and a Ring of Invisibility (legendary) does not even deserve very rare in my opinon.

Rarity is very inconsistent.

1) Frost Brand provides 1d6 extra damage, while Flame Tongue provides 2d6. Both require attunement, and while Frost Brand provides a very nice fire resistance, your offensive items should improve your offense.

2) Turning invisible at will is great, but it takes your action, making your DPR go down in combat.