An instance of the sorting buffer problem consists of a metric space and a server, equipped with a finite-capacity buffer capable of holding a limited number of requests. An additional ingredient of the input is an online sequence of requests, each of which is characterized by a destination in the given metric; whenever a request arrives, it must be stored in the sorting buffer. At any point in time, a currently pending request can be served by drawing it out of the buffer and moving the server to its corresponding destination. The objective is to serve all input requests in a way that minimizes the total distance traveled by the server. In this paper, we focus our attention on instances of the problem in which the underlying metric is either an evenly-spaced or a continuous line metric. Our main findings can be briefly summarized as follows: 1. We present a deterministic O(log n) competitive algorithm for n-point evenly-spaced line metrics. This result improves on a randomized O(log2 n) competitive algorithm due to Khandekar and Pandit. 2. We devise a deterministic O(log N log log N) competitive algorithm for continuous line metrics, where N is the input sequence length. 3. We establish the first non-trivial lower bound for the evenly-spaced case, by proving that the competitive ratio of any deterministic algorithm is at least 2+√3/√3 ≈ 2.154.