In this paper, antisymmetry-constructed clipped optical orthogonal frequency division multiplexing (AC-OFDM) is proposed for visible light communication (VLC) systems, in which an antisymmetry property is imposed directly in time domain. AC-OFDM has nearly the same spectral efficiency and peak-to-average power ratio (PAPR) as traditional asymmetrically clipped optical OFDM (ACO-OFDM) but is less complex to implement. Layered AC-OFDM (LAC-OFDM) is then proposed as an extension to further improve spectral efficiency, where different layers of AC-OFDM signals are added in the time domain and transmitted simultaneously. Computational complexity analysis and numerical results show that LAC-OFDM has nearly the same spectral efficiency as layered asymmetrically clipped optical OFDM (LACO-OFDM) and enhanced unipolar OFDM (eU-OFDM) but is less complex. Specifically, LAC-OFDM requires less than half the multiplication and addition operations compared to the comparable LACO-OFDM scheme. Additionally, a pairwise iterative receiver for LAC-OFDM is proposed and its computational complexity is analysed. Monte Carlo simulation results show that LAC-OFDM requires nearly the same optical signal-to-noise ratio (OSNR) to achieve the same BER performance as LACO- and eU-OFDM.