This paper presents a fast algorithm for the computation of forward and backward sliding conjugate symmetric sequency-ordered complex Hadamard transform (CSSCHT). The forward CSSCHT algorithm calculates the values of window i+N/4 from those of window i and one length-N/4 CSSCHT, one length-N/4 WHT and one length-N/4 modified WHT. The backward CSSCHT algorithm can be obtained by transposing the signal flow graph of that of the forward one. The proposed algorithm requires O(N) arithmetic operations, which is more efficient than the block-based algorithm and those based on the sliding FFT and the sliding DFT. The applications of the sliding CSSCHT in spectrum estimation and transform domain adaptive filtering (TDAF) are also provided with supporting simulation results.