With the massive increase of mobile user and sensor data in the heterogeneous Internet of Things (IoT), the limited caching resources are challenging to meet people's increased data traffic demand. Due to ignoring the influence of uplink, the existing joint caching studies cannot reflect the actual network performance. Furthermore, some emerging delay-sensitive applications in the IoT, such as Virtual Reality, Augmented Reality, and autonomous driving, require better reliability and stability. Digital twins (DT) can provide real-time and stable wireless connectivity for heterogeneous IoT physical and virtual spaces. To address those issues, we propose an adaptive caching scheme based on the evolutionary game to jointly optimize delay and energy consumption in the uplink and downlink of heterogeneous digital twins IoT (HDT-IoT) networks. Besides, we prove the existence of evolutionary stability strategies (ESSs) in the proposed caching scheme. Based on the evolutionary stability analysis results, we derive the related expression between the content popularity and the ESS condition. The numerical results verify the existence of the content evolutionary stability caching strategy and the accuracy of the derived corresponding ESS conditional expressions and evaluate the caching performance by comparing our scheme with the other two caching schemes.