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I've gone through all 27 blocks and noted down what was on each face. The coordinates are counting columns A-I from the top-left to the lower-right of the top view, and top-right to lower-left of the bottom view, and rows from 1-6 from the sphere down.

Within each cube, "Top" and "Bottom" are opposite faces, the two "Left" columns are opposite faces, and the two "Right" columns are opposite faces.

This is the third version of this, with more colors grouped together by the color coding and colors chosen to match the puzzle. The original version can be found here.

I've uploaded my spreadsheet here, as well.

Both the image and the spreadsheet are as accurate as I was able to make it, but note that there may be transcription errors, so double check any conclusions you draw.

Some notes:

• Cube C4 is the only cube without a black sun.
• Cube D3 is the only cube without a white sun.
• Three cubes (B1, G4, I6) have no rings.
• Four cubes (B5, D1, D5, E4) have two sets of rings. (But always never more than three rings total)
• Six cubes (A2, A6, E4, E6, G2, G4) have unusual markings on one face.
• There are exactly 8 solid faces of each of the six non-gray colors, and one face of each color which has markings.
• There are exactly 28 blank faces and 28 faces which have rings at all.

I've gone through all 27 blocks and noted down what was on each face. The coordinates are counting columns A-I from the top-left to the lower-right of the top view, and top-right to lower-left of the bottom view, and rows from 1-6 from the sphere down.

Within each cube, "Top" and "Bottom" are opposite faces, the two "Left" columns are opposite faces, and the two "Right" columns are opposite faces.

This is the third version of this, with more colors grouped together by the color coding and colors chosen to match the puzzle. The original version can be found here.

I've uploaded my spreadsheet here, as well.

Both the image and the spreadsheet are as accurate as I was able to make it, but note that there may be transcription errors, so double check any conclusions you draw.

Some notes:

• Cube C4 is the only cube without a black sun.
• Cube D3 is the only cube without a white sun.
• Seven cubes (A4, D1, E4, E6, G6, H1, H3) have the black and white suns on opposite faces. All the others have them on adjacent faces.
• Three cubes (B1, G4, I6) have no rings.
• Four cubes (B5, D1, D5, E4) have two sets of rings. (But always never more than three rings total)
• Six cubes (A2, A6, E4, E6, G2, G4) have unusual markings on one face.
• There are exactly 8 solid faces of each of the six non-gray colors, and one face of each color which has markings.
• There are exactly 28 blank faces and 28 faces which have rings at all.

Additionally, the 27 cubes correspond to the 27 boxes on each side of the second drawing. Plus, the seven cubes with black and white suns on opposite sides correspond to the seven three-in-a-row sections of the layout. Combining that with the one cube each that only has one sun, I think it's pretty safe to say that the cubes need to be laid out such that each black sun is attached to another cube's white sun, with the two cubes that only have one as the two ends.

Presumably, the symbols on the diagram indicate which face should be up on each cube, and the correct layout will tell us something. But I haven't worked out any correspondence there, yet.

I've gone through all 27 blocks and noted down what was on each face. The coordinates are counting columns A-I from the top-left to the lower-right of the top view, and top-right to lower-left of the bottom view, and rows from 1-6 from the sphere down.

Within each cube, "Top" and "Bottom" are opposite faces, the two "Left" columns are opposite faces, and the two "Right" columns are opposite faces.

This is the second version of this, with more colors grouped together by the color coding. The original version can be found here.

I've uploaded my spreadsheet here, as well.
EDIT: There's a bug in my data. Cell F10 should be Solid Black not Black sun. The notes below are updated to reflect this, but the image and spreadsheet are not. There may be other transcription errors, so double check any conclusions you draw. I also haven't verified that I'm only using the seven colors indicated in the question, so there might be a discrepancy there, too.

Some notes:

• Cube C4 is the only cube without a black sun.
• Cube D3 is the only cube without a white sun.
• Three cubes (B1, G4, I6) have no rings.
• Four cubes (B5, D1, D5, E4) have two sets of rings.
• Six cubes (A2, A6, E4, E6, G2, G4) have unusual markings on one face.

I've gone through all 27 blocks and noted down what was on each face. The coordinates are counting columns A-I from the top-left to the lower-right of the top view, and top-right to lower-left of the bottom view, and rows from 1-6 from the sphere down.

Within each cube, "Top" and "Bottom" are opposite faces, the two "Left" columns are opposite faces, and the two "Right" columns are opposite faces.

This is the third version of this, with more colors grouped together by the color coding and colors chosen to match the puzzle. The original version can be found here.

I've uploaded my spreadsheet here, as well.

Both the image and the spreadsheet are as accurate as I was able to make it, but note that there may be transcription errors, so double check any conclusions you draw.

Some notes:

• Cube C4 is the only cube without a black sun.
• Cube D3 is the only cube without a white sun.
• Three cubes (B1, G4, I6) have no rings.
• Four cubes (B5, D1, D5, E4) have two sets of rings. (But always never more than three rings total)
• Six cubes (A2, A6, E4, E6, G2, G4) have unusual markings on one face.
• There are exactly 8 solid faces of each of the six non-gray colors, and one face of each color which has markings.
• There are exactly 28 blank faces and 28 faces which have rings at all.

I've gone through all 27 blocks and noted down what was on each face. The coordinates are counting columns A-I from the top-left to the lower-right of the top view, and top-right to lower-left of the bottom view, and rows from 1-6 from the sphere down.

Within each cube, "Top" and "Bottom" are opposite faces, the two "Left" columns are opposite faces, and the two "Right" columns are opposite faces.

This is the second version of this, with more colors grouped together by the color coding. The original version can be found here.

I've uploaded my spreadsheet here, as well.

Some notes:

• Cube C4 is the only cube without a black sun.
• Cube C6 is the only cube with two black suns.
• Cube D3 is the only cube without a white sun. (None have two white suns)
• Three cubes (B1, G4, I6) have no rings.
• Four cubes (B5, D1, D5, E4) have two sets of rings.
• Six cubes (A2, A6, E4, E6, G2, G4) have unusual markings on one face.

I've gone through all 27 blocks and noted down what was on each face. The coordinates are counting columns A-I from the top-left to the lower-right of the top view, and top-right to lower-left of the bottom view, and rows from 1-6 from the sphere down.

Within each cube, "Top" and "Bottom" are opposite faces, the two "Left" columns are opposite faces, and the two "Right" columns are opposite faces.

This is the second version of this, with more colors grouped together by the color coding. The original version can be found here.

I've uploaded my spreadsheet here, as well.
EDIT: There's a bug in my data. Cell F10 should be Solid Black not Black sun. The notes below are updated to reflect this, but the image and spreadsheet are not. There may be other transcription errors, so double check any conclusions you draw. I also haven't verified that I'm only using the seven colors indicated in the question, so there might be a discrepancy there, too.

Some notes:

• Cube C4 is the only cube without a black sun.
• Cube D3 is the only cube without a white sun.
• Three cubes (B1, G4, I6) have no rings.
• Four cubes (B5, D1, D5, E4) have two sets of rings.
• Six cubes (A2, A6, E4, E6, G2, G4) have unusual markings on one face.