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We propose a dual-mode (DM) time domain multiplexed (TDM) chirp spread spectrum (CSS) modulation for spectral and energy-efficient low-power wide-area networks (LPWANs). DM-CSS modulation that uses both the even and odd cyclic time shifts has been proposed for LPWANs to achieve noteworthy performance improvement over classical counterparts. However, its spectral efficiency (SE) is half of the in-phase and quadrature (IQ)-TDM-CSS scheme that employs IQ components with both up and down chirps, resulting in a SE that is four times relative to Long Range (LoRa) modulation. Nevertheless, the IQ-TDM-CSS scheme only allows coherent detection. Furthermore, it is also sensitive to carrier frequency and phase offsets, making it less practical for low-cost battery-powered LPWANs for Internet-of-Things (IoT) applications. DM-CSS uses either an up-chirp or a down-chirp. DM-TDM-CSS consists of two chirped symbols that are multiplexed in the time domain. One of these symbols consisting of even and odd frequency shifts (FSs) is chirped using an up-chirp. The second chirped symbol also consists of even and odd FSs, but they are chirped using a down-chirp. It shall be demonstrated that DM-TDM-CSS attains a maximum achievable SE close to IQ-TDM-CSS while also allowing both coherent and non-coherent detection. Additionally, unlike IQ-TDM-CSS, DM-TDM-CSS is robust against carrier frequency and phase offsets.