There are several mnemonics for the difference between a Galeazzi and a Monteggia fracture-dislocation:
GRIMUS
MUGR (pronounced as mugger)
FROG
GRUesome MURder
Manchester United / Glasgow Rangers
Monteggia is mountainous
It is useful to note that it is the head of the non-fractured bone that is dislocated.
An alternative to the various mnemonics is the following simple rule set for eponymous fracture-dislocations of the forearm:
MonteggiA rhymes with ULNA (emphasis "ah" sound). It is the only one of the eponymous forearm fracture-dislocations to involve the ulna. A being the first letter of the alphabet, there is dislocation of the proximal radioulnar joint.
Galeazzi and Essex-Lopresti fractures are the inverse; both involve the radius with dislocation of the distal radioulnar joint. Essex-Lopresti starts with E for elbow; indicating that it involves the radial head or neck wheras Galeazzi involves the radial diaphysis.
Mnemonics
GRIMUS
GRIMUS helps to remember which forearm bone is fractured and whether the distal ("inferior") or proximal ("superior") part of the other forearm bone is dislocated.
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G: Galeazzi
R: radius
I: inferior
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M: Monteggia
U: ulna
S: superior
MUGR
Monteggia involves fracture of the ulna with proximal radial dislocation
Galeazzi involves fracture of the radius with distal radioulnar dislocation
FROG
Fractured radius of Galeazzi
GRUesome MURder
GRUesome MURder assists remembering which bone is firstly fractured and then secondly which is dislocated:
G: Galeazzi R: radius fracture U: ulna dislocation
M: Monteggia U: ulna fracture R: radial head dislocation
Manchester United / Glasgow Rangers
Two British football (soccer) teams, with devoted followings:
Man U: Monteggia / ulnar fracture
Glasgow Rangers: Galeazzi / radial fracture
Monteggia is mountainous
Monteggia sounds close to mountain and since a mountain is high up, Monteggia is more proximal (higher) than Galeazzi.