Another option is to use a tick function. You can also add formatting to the tick labels inside this tick function if needed. See NumberForm and related functions.

      tick[min_, max_] := Table[{i, Abs@i}, {i, Ceiling[min], Floor[max], 0.5}]
      SphericalPlot3D[1 + 2 Cos[2 \[Theta]], {\[Theta], 0, Pi}, {\[Phi], 0, 2 Pi}, Ticks -> tick]

You have to set the Ticks option manually like this:

SphericalPlot3D[1+2Cos[2\[Theta]],{\[Theta],0,Pi},{\[Phi],0,2Pi},Ticks->{{#,Abs[#]}&/@Range[-1,1,0.5],{#,Abs[#]}&/@Range[-1,1,0.5],{#,Abs[#]}&/@Range[-6,6,0.5]}]

or equivalently:

SphericalPlot3D[1+2Cos[2\[Theta]],{\[Theta],0,Pi},{\[Phi],0,2Pi},Ticks->{{{-1.`,1.`},{-0.5`,0.5`},{0.`,0.`},{0.5`,0.5`},{1.`,1.`}},{{-1.`,1.`},{-0.5`,0.5`},{0.`,0.`},{0.5`,0.5`},{1.`,1.`}},{{-6.`,6.`},{-5.5`,5.5`},{-5.`,5.`},{-4.5`,4.5`},{-4.`,4.`},{-3.5`,3.5`},{-3.`,3.`},{-2.5`,2.5`},{-2.`,2.`},{-1.5`,1.5`},{-1.`,1.`},{-0.5`,0.5`},{0.`,0.`},{0.5`,0.5`},{1.`,1.`},{1.5`,1.5`},{2.`,2.`},{2.5`,2.5`},{3.`,3.`},{3.5`,3.5`},{4.`,4.`},{4.5`,4.5`},{5.`,5.`},{5.5`,5.5`},{6.`,6.`}}}]