This diagram (without the red line) comes from a paper by Edward Hacker and Steve Moore – 'A brief note on the two-part division of the received order of the hexagrams in the Zhouyi' published in the June 2003 issue of the Journal of Chinese Philosophy (30:2, pp 219–221). They discovered a diagram showing their idea was known in the Yuan (Mongol) dynasty. See http://biroco.com/division.htm
It seems to solve the mystery of the division of the hexagrams in two unequal parts (the 'upper' canon of 30 and 'lower' of 34 hexagrams). The hexagrams have the hexagram name written at one side, and the name of its inverted one upside down on the other side (I rotated the diagram 90°). So a hexagram, which does not change when inverted, should have on both sides the same name. In the diagram these double names are left blank (eg lower left corner).
For the 'mirrors' or line pathways it explains the relation of e.g. hex.3 line 2 with hex.4 line 5 (third hex. at left) and with hex.59 line 5 and 60 line 2. Hex.3 line 2 is fan-yao of hex.60 line 2 (when 3.2 changes, 60.2 corresponds with 3.2), and qian-yao with 4 line 5 (hex.3 upside down is hex.4, line 3.2 becomes 4.5).
To find the Mirror for a specific line: go to its hexagram, and click on the Mirror of this line.
An example, Mirror 33.2-44.2-43.5-34.5
Lines 2 and 5 have to do with relations. To people, to objects, to needs, to actions, to circumstances. Here it is about your relation to possessions, or the "way to possess". To get grip on it (43.5); hold it fast (33.2); leave it free (44.2); let it go (34.5).
What about hexagrams which do not change when upside down, like hex.2:
The paired hexagram of hex.2 is hex.2, because upside down is it still hex.2. What changes is the line: now it is line 4.
This is an image of the line pathway starting at hex.2 line 3. When the image is rotated 90° counter-clockwise, hex.2 is at the bottom left, with line 3 changing. Next to it its fanyao 15.3. Rotated 90° clockwise hex.2 is at bottom right, with next to it its fanyao 16.4.
this is Mirror 2.3-15.3-2.4-16.4
Top left is hex.2 line 3 changing. Bottom left is its fanyao: hex.15.3.
Top right the qianyao of hex.2: its paired hexagram, which is again hex.2, but now with line 4 changing. Bottom right the fanyao of 2.4, which is 16.4.
Every Mirror has a theme, depending on the lines and hexagrams involved. In the first example above, 33.2, the theme was how to deal with "possessing".
The theme of the mirror of 2.3 is, that everything essential you do comes from your heart, not from what you want. Wanting is limited and gives small results, following your heart is big.
Genuine motivation attracts others(16.4)
you act from out that germ, not towards a result (2.3)
in coöperation, you give all of yourself (15.3)
when nothing stirs inside, you are silent (2.4)