This work proposes a new simultaneous wireless power transfer and modulation classification (SWPTMC) scheme appropriate for internet of things (IoT) applications. The problem of SWPTMC is investigated for various modulation formats, i.e., quadrature phase-shift-keying (QPSK), π/4-QPSK, offset QPSK (OQPSK), 16-pulse amplitude modulation (16-PAM), 16-quadrature amplitude modulation (16-QAM), and minimum shift keying (MSK). We propose three receiver architectures, i.e., an integrated receiver, a separate receiver with power splitting (PS), and a separate receiver with energy harvesting (EH)-based classification; all the architectures are studied under a non-linear model with a certain sensitivity and saturation level. Also, we derive the average harvested power over a Rayleigh fading channel for the different modulation formats. Two different approaches are used for the blind modulation classification (MC) algorithm: one for the intermediate frequency signal and the other for the baseband signal. Both the MC algorithms are based on the higher-order cumulants and cyclic cumulants of the received signal. The cyclic cumulants use the non-zero cycle frequency position, while the cumulants use threshold values for classifying modulation formats. Monte Carlo simulations are used to evaluate the performance of the proposed SWPTMC schemes. The results show that we can simultaneously harvest power without much affecting the classifier performance. Moreover, with an integrated receiver, we can simultaneously perform MC and harvest power without the requirement of PS circuit.