We introduce a new approach for computing the monodromy of the Hitchin map and use this to completely determine the monodromy for the moduli spaces of
L
L
-twisted
G
G
-Higgs bundles for the groups
G
=
G
L
(
2
,
C
)
G = GL(2,\mathbb {C})
,
S
L
(
2
,
C
)
SL(2,\mathbb {C})
, and
P
S
L
(
2
,
C
)
PSL(2,\mathbb {C})
. We also determine the Tate-Shafarevich class of the abelian torsor defined by the regular locus, which obstructs the existence of a section of the moduli space of
L
L
-twisted Higgs bundles of rank
2
2
and degree
deg
(
L
)
+
1
\deg (L)+1
. By counting orbits of the monodromy action with
Z
2
\mathbb {Z}_2
-coefficients, we obtain in a unified manner the number of components of the character varieties for the real groups
G
=
G
L
(
2
,
R
)
G = GL(2,\mathbb {R})
,
S
L
(
2
,
R
)
SL(2,\mathbb {R})
,
P
G
L
(
2
,
R
)
PGL(2,\mathbb {R})
,
P
S
L
(
2
,
R
)
PSL(2,\mathbb {R})
, as well as the number of components of the
S
p
(
4
,
R
)
Sp(4,\mathbb {R})
and
S
O
0
(
2
,
3
)
SO_0(2,3)
-character varieties with maximal Toledo invariant. We also use our results for
G
L
(
2
,
R
)
GL(2,\mathbb {R})
to compute the monodromy of the
S
O
(
2
,
2
)
SO(2,2)
Hitchin map and determine the components of the
S
O
(
2
,
2
)
SO(2,2)
character variety.