We explore the phenomenon commonly known as halo assembly bias, whereby dark matter halos of the same mass are found to be more or less clustered when a second halo property is considered, for halos in the mass range 3.7×1011h−1M⊙−5.0×1013h−1M⊙. Using the Large Suite of Dark Matter Simulations (LasDamas) we consider nine commonly used halo properties and find that a clustering bias exists if halos are binned by mass or by any other halo property. This secondary bias implies that no single halo property encompasses all the spatial clustering information of the halo population. The mean values of some halo properties depend on their halo's distance to a more massive neighbor. Halo samples selected by having high values of one of these properties therefore inherit a neighbor bias such that they are much more likely to be close to a much more massive neighbor. This neighbor bias largely accounts for the secondary bias seen in halos binned by mass and split by concentration or age. However, halos binned by other mass-like properties still show a secondary bias even when the neighbor bias is removed. The secondary bias of halos selected by their spin behaves differently than that for other halo properties, suggesting that the origin of the spin bias is different than of other secondary biases.