The palindromic nature of a DNA sequence can be known even in single stranded condition because

First half of palindromic sequence is complementary to second half. The palindrome in DNA is a sequence of base pairs that reads same on the two strands when orientation of reading is kept the same. For example, the following sequences reads the same on the two strands in 5' →3' direction. This is also true if read in the 3' →  5' direction.