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My favorite being a slight variation of @Lou's own solution

Transpose[{data[[;;, 1, 1]], data[[;;, -1, -1]]}]

you can also use

Transpose@Rest@Extract[data, {{}, {;; , 1, 1}, {;; , -1, -1}}]

or

ReplacePart[data, p_ :> {data[[p, 1, 1]], data[[p, -1, -1]]}]

or

{First@First@#, Last@Last@#} & /@ data

or

ClearAll[f];
f[{{a_, __}, {__, b_}}] := {a, b}
f /@ data

all give

My favorite being a slight variation of @Lou's own solution

Transpose[{data[[;;, 1, 1]], data[[;;, -1, -1]]}]

you can also use

Transpose@Rest@Extract[data, {{}, {;; , 1, 1}, {;; , -1, -1}}]
(* version 9 only -- does not work in version 10 -- thanks: Simon Woods *)

or

ReplacePart[data, p_ :> {data[[p, 1, 1]], data[[p, -1, -1]]}]

or

{First@First@#, Last@Last@#} & /@ data

or

ClearAll[f];
f[{{a_, __}, {__, b_}}] := {a, b}
f /@ data

all give

Transpose@Rest@Extract[data, {{}, {;; , 1, 1}, {;; , -1, -1}}]

or

ReplacePart[data, p_ :> {data[[p, 1, 1]], data[[p, -1, -1]]}]

or

{First@First@#, Last@Last@#} & /@ data

or

ClearAll[f];
f[{{a_, __}, {__, b_}}] := {a, b}
f /@ data

all give

My favorite being a slight variation of @Lou's own solution

Transpose[{data[[;;, 1, 1]], data[[;;, -1, -1]]}]

you can also use

Transpose@Rest@Extract[data, {{}, {;; , 1, 1}, {;; , -1, -1}}]

or

ReplacePart[data, p_ :> {data[[p, 1, 1]], data[[p, -1, -1]]}]

or

{First@First@#, Last@Last@#} & /@ data

or

ClearAll[f];
f[{{a_, __}, {__, b_}}] := {a, b}
f /@ data

all give

{First@First@#, Last@Last@#} & /@ data

or

ClearAll[f];
f[{{a_, __}, {__, b_}}] := {a, b}
f /@ data

Transpose@Rest@Extract[data, {{}, {;; , 1, 1}, {;; , -1, -1}}]

or

ReplacePart[data, p_ :> {data[[p, 1, 1]], data[[p, -1, -1]]}]

or

{First@First@#, Last@Last@#} & /@ data

or

ClearAll[f];
f[{{a_, __}, {__, b_}}] := {a, b}
f /@ data

all give