We study neural network models that learn location invariant orthographic representations for printed words. We compare two model architectures: with and without a hidden layer. We find that both architectures succeed in learning the training data and in capturing benchmark phenomena of skilled reading transposed-letter and relative-position priming. Networks without a hidden layer use a strategy for identifying target words based on the presence of letters in the target word, but where letter contributions are modulated using the interaction between within-word position and within-slot location. This modulation allows networks to factor in some information about letter position, which is sufficient to segregate most anagrams. The hidden layer appears critical for success in a lexical decision task, i.e., sorting words from non-words. Networks with a hidden layer better succeed at correctly rejecting non-words than networks without a hidden layer. The latter tend to over-generalize and confuse non-words for words that share letters.