Convert column suffixes from pandas join into a MultiIndex

You can use pd.concat with the keys argument, along the first axis:

df = pd.concat([df_L, df_R], keys=['L','R'],axis=1).swaplevel(0,1,axis=1).sort_index(level=0, axis=1)

>>> df
   X     Y   
   L  R  L  R
0  1  2  5  6
1  3  4  7  8