Deleting the same lines from a list

DeleteDuplicatesBy[lines, Sort @@ # &]

DeleteDuplicates[Sort @@@ lines]

The key is to realize that we have to Apply Sort one level deeper because of the head Line.

Another possibility is to use RegionEqual:

Length @ lines
Length @ DeleteDuplicates[lines, RegionEqual]

8

7

If your lines had more than 2 segments, than using Sort to canonicalize wouldn't work.

It is because the arguments of the 5-th and 8-th terms go in different order. Indeed, let us take only the arguments:

arg = lines[[All, 1]];

and let us try to compare the arguments of the 5th and 8th terms:

arg[[5]] === arg[[8]]

(*  False  *)

Now let us sort the arguments:

newArg = Sort /@ lines[[All, 1]]

and let us check the 5th and 8th terms

newArg[[5]] === newArg[[8]]

(*  True  *)

Now the duplicates will be deleted, and here is the result:

Line /@ DeleteDuplicates[newArg]

Have fun!

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