05AB1E, 2 bytes

(1

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!enilno ti yrT

( # negate nothing
 1 # push 1 (and implictly output it)

1 # push 1
 ( # negate it (and implictly output it)

Whitespace, 21 bytes

S S S T N
T N
S T N
N
N
T S N
T N
T T S S 

Letters S (space), T (tab), and N (new-line) added as highlighting only.

Outputs 1/-1.

Try it online or try it online reversed (with raw spaces, tabs and new-lines only).

Explanation:

Utilizing the Exit Program builtin being a short palindrome NNN.

The regular program will:

SSSTN # Push 1 to the stack
TNST # Pop and print the top of the stack as number
NNN # Exit the program, making everything after it no-ops

The reverse program will:

SSTTN # Push -1 to the stack
TNST # Pop and print the top of the stack as number
NNN # Exit the program, making everything after it no-ops

Small additional explanation of pushing a number:

• First S: Enable Stack Manipulation


• Second S: Push a number to the stack


• 
  S or T: Positive/negative respectively


• Some S/T followed by a trailing N: number in binary, where S=0 and T=1
  

I.e. SSTTSTSN pushes -10.

JavaScript (V8), 19 bytes

print(1)//)1-(tnirp

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almost identical to...

C# (Visual C# Interactive Compiler), 19 bytes

Print(1)//)1-(tnirP

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(thanks to @someone for pointing it out)

but shorter in...

Python 2, 15 bytes

print 1#1-tnirp

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Even shorter in PHP, because it has this magic printing tool: <?= ...

PHP, 12 bytes

<?=1;#;1-=?<

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Runic Enchantments, 4 bytes

1@Z1

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I couldn't find a way to re-use the 1 command, even at the expense of a byte or two.

1@ɩ also works, but is the same number of bytes.

Klein 011, 5 bytes

1-
@/

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Reversed

/@
-1

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These take advantage of Klein's unique topology, specifically the real projective plane. (Although individually each answer only needs a Klein bottle).

///, 4 bytes

9/9-

Outputs 9.

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Reversed:

-9/9

Outputs -9.

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Everything before the / is printed, while the rest is ignored (not really used slashes much so I don't know exactly what happens, but it doesn't output anything).

Japt, 2 bytes

Any single digit integer >0 can be used in place of the 2 as can A-G, H, I, J or L (10-16, 32, 64, -1 & 100, respectively).

n2

Test it | Reversed

The n method when applied to an integer, subtracts that integer from the argument passed to it, which defaults to 0. When run forwards, the n method is being run on the implicit first input, which also defaults to 0.

Alternatively, the g method could be used instead of n, which gives the sign of the result of subtracting its argument from the integer it's applied to.

PowerShell,  4  3 bytes

1-0

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Golfed a byte by using arithmetic instead of the number-comment-number format.

Cubix, 7 6 bytes

@)RO(O

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Reversed

Explanation

Cubified:

 @
) R O (
 O

Unrolling the control flow, we execute )O(@, which increments, outputs, decrements, and exits.

Reversed and cubified:

 O
( O R )
 @

Unrolling the control flow, we execute (O@, which decrements, outputs, and exits.

Previous version

@O(.)O@

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Reversed

Not as short, but much more aesthetically pleasing.

Jelly, 2 bytes

NC

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Negative, results in 0, then Complement, results in 1-0 = 1.

CN

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Complement, results in 1-0 = 1. Negative, results in -1.

APL (Dyalog Unicode),  4  3 bytes

1-0

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Trivial answer. Prints 1 when run and ¯1 when run reversed.

Haskell, 28 bytes

main=print 1--)1-(tnirp=niam

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T-SQL, 16 bytes

--Forwards:
PRINT 4--4-TNIRP

--Backwards:
PRINT-4--4 TNIRP

Picked 4 because 1 is overused :)

Triangular, 4 bytes

i%%d

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Outputs 1.

Ungolfed:

 i
 % %
 d

Reversed:

d%%i

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Outputs -1.

MATL, 3 bytes

Nqv

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• Normal


• Reversed

How they work

Normal:

N % Push number of elements in the stack: 0
q % Subtract 1: gives -1
v % Concatenate stack contents vertically: leaves -1 as is
 % Implicit display stack contents

Reversed:

v % Concatenate stack contents vertically: gives the empty array, []
q % Subtract 1: leaves [] as is
N % Push number of elements in the stack: 1
 % Implicit display. [] is not displayed

MarioLANG, 5 bytes

+
:
-

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Reversed:

-
:
+

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Ruby, 8 7 bytes

p 1#0~p

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Thanks Arnauld for chipping off 1 byte and making it less dull.

Java 5 or 6, 127 67 bytes

enum AA;System.out.print(9);//}};)9-(tnirp.tuo.metsyS;AA mune

Outputs 9/-9.

No online compiler, because Java 5 or 6 isn't available anywhere.

You can however try this 127 bytes Java 8 equivalent:
Try it online or try it online reversed.

Explanation:

enum A // Create an enum
 A; // With a mandatory value
 // And in a separate instance code-block:
 System.out.print(9); // Print 9 to STDOUT
//;)9-(tnirp.tuo.metsyS;AA mune // Comment and thus a no-op

Java 5 and 6 had a bug allowing you to create a code block inside an enum to do something, despite missing a program's mandatory main-method. This will result in an error:

java.lang.NoSuchMethodError: main

Exception in thread "main"

But will still output what we'd want to STDOUT first, so we can ignore that.