Interdependent controls in Manipulate

Not sure if this is the best way, but you could consider something like this:

Manipulate[
 {x, y},
 {x, Manipulator[Dynamic[x, (x = #; y = 1/#) &], {.1, 10}] &},
 {y, Manipulator[Dynamic[y, (y = #; x = 1/#) &], {.1, 10}] &},
 Initialization :> ({x, y} = {1, 1})]

Edit V10

Since V10 one can use a shorter form:

Manipulate[
    {x, y}
  , {x, .1, 10, TrackingFunction :> ((x = #; y = 1/#) &)}
  , {y, .1, 10, TrackingFunction :> ((y = #; x = 1/#) &)}
  , Initialization :> ({x, y} = {1, 1})
]

You could also build a custom Manipulate-like object using sliders

Panel[DynamicModule[{x, y},
    Column[{
        Grid[{
            {"x", Slider[Dynamic[x, (x = #; y = 1/#) &], {0.1, 10}], Dynamic[x]},
            {"y", Slider[Dynamic[y, (y = #; x = 1/#) &], {0.1, 10}], Dynamic[y]}
        }],
        Panel[{Dynamic[x], Dynamic[y]}, ImageSize -> 300, Background -> White]
    }], ImageSize -> 300
]]

This?

{Slider[Dynamic[x], {0.1, 10}], 
 Slider[Dynamic[1/ x, Set[x, 1/#] &], {0.1, 10}]}
Dynamic[x]