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I see you as a circle.
You see me as a point.
And when you need direction
I never disappoint.

You see me when you're sleeping,
And oft when you awake
(Even in the daytime
I sit and burn and bake).

Your friends and you are whirling.
Is that really you?
Yes! Of course! I recognize
That swirling bit of blue.

Who are you and I?

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Is it

Polaris, the North Star (I) (might be a different star used for navigation) and the Earth (you)

I see you as a circle. You see me as a point. (unsure)

The earth is typically drawn as a circle, and stars as points. This is the most shaky clue, as from the star's distance the Earth would be invisibly tiny so not really a circle)

And when you need direction I never disappoint.

Polaris is famously used for navigation, as it doesn't move in the sky

You see me when you're sleeping, And oft when you awake

Stars are visible at night. They are occasionally visible during the day during a solar eclipse

(Even in the daytime I sit and burn and bake).

Stars still shine during the day, it's just too bright to see them.

Your friends and you are whirling. Is that really you?

Planets in the solar system spin both on their axes and around the sun

Yes! Of course! I recognize That swirling bit of blue.

Earth is known as the blue planet, and the clouds are swirling in photos from space

Anagram for Umbrella, misspelled

From JS1 in the comments: polaris is an anagram of parisol(parasol), which is a type of umbrella

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    $\begingroup$ Rot13(Nyfb, cbynevf vf na nantenz sbe "cnevfby", juvpu vf cnenfby (na hzoeryyn) zvffcryyrq.) $\endgroup$ – JS1 Dec 19 '19 at 4:02
  • $\begingroup$ Oh, that's perfect! I was trying to find anagrams of umbrella itself and couldn't come up with anything. $\endgroup$ – Amy Dec 19 '19 at 4:12
  • $\begingroup$ I (the point), do not move because I'm rot13(qverpgyl nobir lbhe Abegu cbyr). The circle is because, most likely, it is not in such alignment when viewed by "me," rot13(vg ebgngrf guebhtu gur fxl). $\endgroup$ – Chowzen Dec 19 '19 at 10:24
  • $\begingroup$ Very nicely done! I'll award the green ✓ after I read my horoscope. :) $\endgroup$ – Chowzen Dec 19 '19 at 10:26

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