Reflection modulation based on reconfigurable intelligent surface (RIS) is considered to be a promising information transfer mechanism, which does not require any additional radio frequency chains. However, existing reflection modulation schemes consider manipulating the ON/OFF states of RIS elements, which may result in some power loss. In this paper, we propose a new scheme, called RIS-aided quadrature reflection modulation (RIS-QRM), for better utilizing the reflection power in RIS-aided downlink multiple-input single-output (MISO) wireless systems. To this end, RIS-QRM partitions the RIS elements into two subsets in order to reflected the incident signals towards two orthogonal directions by using passive beamforming and encodes its local data onto the two partitions. A closed-form expression for the unconditional pairwise error probability of RIS-QRM in Rician fading is derived assuming maximum-likelihood detection at the receiver. Moreover, a low-complexity detection method that decouples the joint search of the constellation symbol and the RIS element partition is proposed. Computer simulation results corroborate the effectiveness of RIS-QRM and validate the analytical results. It is shown that RIS-QRM improves the error performance of the additional bits delivered by the RIS without deteriorating the error performance of the bits modulated on the constellation symbol, as compared to ON/OFF-based RIS-aided schemes.