As I have written in a comment, do never use list (or any built-in) as a variable name:
L = [['1', '2'], ['1', '3'], ['1', '4'], ['1', '5'], ['2', '1'], ['4', '1'], ['2', '6']]
Have a look at your code:
for i in range(len(L)):
    if L[i] == L[i][::-1]:
        print("Match found")
        del L[i][::-1]
There are two issues. First, you compare L[i] with L[i][::-1], but you want to compare L[i] with L[j][::-1] for any j != i. Second, you try to delete elements of a list during an iteration. If you delete an element, then the list length is decreased and the index of the loop will be out of the bounds of list:
>>> L = [1,2,3]
>>> for i in range(len(L)):
...     del L[i]
... 
Traceback (most recent call last):
...
IndexError: list assignment index out of range
To fix the first issue, you can iterate twice over the elements: for each element, is there another element that is the reverse of the first? To fix the second issue, you have two options: 1. build a new list; 2. proceed in reverse order, to delete first the last indices.
First version:
new_L = []
for i in range(len(L)):
    for j in range(i+1, len(L)):
        if L[i] == L[j][::-1]:
            print("Match found")
            break
    else: # no break
        new_L.append(L[i])
print(new_L)    
Second version:
for i in range(len(L)-1, -1, -1):
    for j in range(0, i):
        if L[i] == L[j][::-1]:
            print("Match found")
            del L[i]
print(L)    
(For a better time complexity, see @yatu's answer.)
For a one-liner, you can use the functools module:
>>> L = [['1', '2'], ['1', '3'], ['1', '4'], ['1', '5'], ['2', '1'], ['4', '1'], ['2', '6']]
>>> import functools
>>> functools.reduce(lambda acc, x: acc if x[::-1] in acc else acc + [x], L, [])
[['1', '2'], ['1', '3'], ['1', '4'], ['1', '5'], ['2', '6']]
The logic is the same as the logic of the first version.