How does a Cloak of Displacement interact with truesight/blindsight?

Question

Cloak of Displacement:

While you wear this cloak, it projects an illusion that makes you appear to be standing in a place near your actual location, causing any creature to have disadvantage on attack rolls against you. If you take damage, the property ceases to function until the start of your next turn. This property is suppressed while you are incapacitated, restrained, or otherwise unable to move.

Truesight:

A monster with truesight can, out to a specific range, see in normal and magical darkness, see invisible creatures and objects, automatically detect visual illusions and succeed on saving throws against them, and perceive the original form of a shapechanger or a creature that is transformed by magic. Furthermore, the monster can see into the Ethereal Plane within the same range.

Blindsight:

A monster with blindsight can perceive its surroundings without relying on sight, within a specific radius. Creatures without eyes, such as grimlocks and gray oozes, typically have this special sense, as do creatures with echolocation or heightened senses, such as bats and true dragons. If a monster is naturally blind, it has a parenthetical note to this effect, indicating that the radius of its blindsight defines the maximum range of its perception.

Illusion spells:

Illusion spells deceive the senses or minds of others. They cause people to see things that are not there, to miss things that are there, to hear phantom noises, or to remember things that never happened. Some illusions create phantom images that any creature can see, but the most insidious illusions plant an image directly in the mind of a creature.

I guess it all depends on whether the illusory effect of the Cloak of Displacement projects a visual illusion (appears to be the case) that can be seen through with the extra senses or the illusion is on the minds of those around. Any specific rules or lore that I am missing?

Answer 1

Both Truesight and Blindsight see through the illusion

The cloak of displacement is not one of the most insidious illusions that plant an image directly in the mind of a creature. It projects an illusion to another spot. Since that illusionary image fools sight, both of these features are immune to it.

The cloak of displacement is traditionally made from the hide of Displacer Beasts (as you can also see in the image, which even still has the Displacer Beast tentacles attached). The Displacer Beasts displacement is a visual effect according to its lore entry in the Monster Manual:

This monstrous predator takes its name from its ability to displace light so that it appears to be several feet away from its actual location.

While the cloak does not state it does the same thing, it is a good indicator this is how it works.

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As an aside, the kinds of illusions that create something in a creature's mind typically will tell you so and often are called phantasms. For example the spell phantasmal force says:

You craft an illusion that takes root in the mind of a creature that you can see within range.

Answer 2

Truesight ignores the cloak, Blindsight does not

The cloak “projects an illusion” and illusion’s “deceive the senses or minds of others”.

It isn’t deceiving the mind because illusions that do that are explicit that that’s what they do.

So, it’s deceiving the senses - all the senses; because illusions that deceive only one sense are specific that that’s what they do.

Truesight

The illusion is partially visible so that part is ignored by Truesight. Even though the other senses are still deceived we can, for visually dominated creatures (i.e. most creatures, especially if they have Truesight), assume that being able to clearly see the correct location negates the advantage.

Blindsight

Whatever sense the creature uses for their Blindsight is being deceived, so they have disadvantage.