This paper considers the context of orthogonal space-time block coded OFDM (OSTBC-OFDM) without channel state information at the receiver. Assuming noncoherent maximum-likelihood detection, the interest herein lies in detection within one OSTBC-OFDM block, motivated by its capability of accommodating relatively fast block fading channels. Our investigation focuses on analysis aspects, where we seek to establish practical noncoherent BPSK/QPSK OSTBC-OFDM schemes that have provably good channel identifiability and diversity properties. We consider perfect channel identifiability (PCI), a strong condition guaranteeing unique noncoherent channel identification for any (nonzero) channel. Through a judicious design involving special OSTBCs and pilot placement, we propose an OSTBC-OFDM scheme that is PCI-achieving and consumes fewer pilots compared to conventional pilot-aided channel estimation methods. We further our analysis by showing that a PCI-achieving scheme also achieves maximal noncoherent spatial diversity for the Kronecker Gaussian spatial-temporal channel fading model, which covers the popular i.i.d. Rayleigh fading channel and a variety of correlated and sparse multipath channels. All these results are developed in parallel for the centralized point-to-point MIMO scenario and a distributed relay communication scenario. For the latter scenario, our diversity analysis shows that the PCI-achieving scheme can also achieve maximal noncoherent cooperative diversity. The performance merits of the proposed PCI-achieving scheme are demonstrated by simulations.