You can do the following:
df =DataFrame({'a':[1,2,3,4],'b':[2,4,6,8]})
df['x']=df.a + df.b
df['y']=df.a - df.b
create column title whatever order you want in this way:
column_titles = ['x','y','a','b']
df.reindex(columns=column_titles)
This will give you desired output