What are all the blocks that can face a specific direction in vanilla Minecraft?

In Minecraft, block orientation can be saved as a block's metadata value or some sort of additional data inside a tile entity. There is no standard method to do so, though most blocks use some bits of their metadata value for their orientation. Some of them have different orientations, but it is calculated automatically.

A full list of such data is available at the Minecraft wiki in the "Data Values" article. From the article, the following blocks can have a specific orientation:

• flowing water and lava (the orientation is automatically calculated by the game). Stationary water and lava have no orientation.
• Anvils (orientation in the lowest bit of the metadata value)
• Beds (orientation in the lowest two bits of the metadata value)
• Buttons (orientation in the lowest two bits of the metadata value)
• Chests (orientation in the metadata value)
• Cocoas (orientation in the lowest two bits of the metadata value)
• Dispensers (orientation in the lowest three bits of the metadata value)
• Doors (orientation in the lowest two bits of the metadata value of the bottom part and the lowest one bit of the upper part)
• Droppers (orientation in the lowest three bits of the metadata value)
• End Portal Blocks (orientation in the lowest two bits of the metadata value)
• Fence Gates (orientation in the lowest two bits of the metadata value)
• Fences (orientation calculated automatically)
• Furnaces (orientation in the metadata value)
• Glass Panes (orientation calculated automatically)
• Hay Bales (orientation in the metadata value)
• Heads (orientation in the metadata value if on the wall, in the tile entity if on the floor)
• Hoppers (orientation in the lowest three bits of the metadata value)
• Huge Mushroom blocks (orientation in the metadata value)
• Iron Bars (orientation calculated automatically)
• Ladders (orientation in the metadata value)
• Levers (orientation in the lowest three bits of the metadata value)
• Pillar Blocks of Quartz (orientation in the metadata value; the three possible ones correspond to values 2, 3 and 4; values of 0 and 1 create non-orientated different blocks of quartz)
• Pistons and Piston Extensions (orientation in the lowest three bits of the metadata value)
• Pumpkins and Jack'o'Lanterns (orientation in the lowest two bits of the metadata value, with a special case of metadata = 4 meaning "no face/orientation")
• Pumpkin Stems and Melon Stems (orientation automatically calculated)
• Rails (orientation in the lowest three bits of the metadata value for special rails, in all the bits of the metadata value for regular rails - that's why powered rails are always straight or angled, never making a turn)
• Redstone Repeaters (orientation in the lowest two bits of the metadata value)
• Redstone Wire (orientation calculated at runtime)
• Single Slabs orientation - upper or lower part of a block - in the highest bit of the metadata value)
• free-standing Sign Posts (orientation in the metadata value)
• Stairs (orientation partially in the metadata value; connecting "corner" stairs' orientation calculated automatically)
• Torches and Redstone Torches (orientation in the metadata value)
• Trapdoors (orientation in the lowest two bits of the metadata value)
• Tripwire (orientation automatically calculated)
• Tripwire Hooks (orientation in the lowest two bits of the metadata value)
• Vines (orientation in the metadata value)
• Wall Signs (orientation in the metadata value)
• Wood (orientation in the highest-order two bits of the metadata value)

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Answer written for version: 1.6.2