In split inference, a deep neural network (DNN) is partitioned to run the early part of the DNN at the edge and the later part of the DNN in the cloud. This meets two key requirements for on-device machine learning: input privacy and computation efficiency. Still, an open question in split inference is output privacy, given that the outputs of the DNN are observable in the cloud. While encrypted computing can protect output privacy too, homomorphic encryption requires substantial computation and communication resources from both edge and cloud devices. In this paper, we introduce Salted DNNs: a novel approach that enables clients at the edge, who run the early part of the DNN, to control the semantic interpretation of the DNN's outputs at inference time. Our proposed Salted DNNs maintain classification accuracy and computation efficiency very close to the standard DNN counterparts. Experimental evaluations conducted on both images and wearable sensor data demonstrate that Salted DNNs attain classification accuracy very close to standard DNNs, particularly when the Salted Layer is positioned within the early part to meet the requirements of split inference. Our approach is general and can be applied to various types of DNNs. As a benchmark for future studies, we open-source our code.