When should I use a bitwise operator?

Bitwise is useful for things in PHP just like anything else.

How about a value that can have multiple states turned on at the same time?

<?php

// since we're setting constant values in base10 we must progressively double
// them since bitwise operations work in base2. you'll see why when we output
// these as binary values below.
const STATE_FOO = 1;
const STATE_BAR = 2;
const STATE_FEZ = 4;
const STATE_BAZ = 8;

// show base2 values of the above constants
echo sprintf("STATE_FOO's base2 value is %08d\n", decbin(STATE_FOO));
echo sprintf("STATE_BAR's base2 value is %08d\n", decbin(STATE_BAR));
echo sprintf("STATE_FEZ's base2 value is %08d\n", decbin(STATE_FEZ));
echo sprintf("STATE_BAZ's base2 value is %08d\n\n", decbin(STATE_BAZ));

// set state to FOO and FEZ
$state = STATE_FOO | STATE_FEZ;

echo sprintf("base10 value of \$state is %s\n", $state);
echo sprintf("base2 value of \$state is %08d\n", decbin($state));
echo sprintf("Does \$state include FOO state? %s\n", (bool)($state & STATE_FOO));
echo sprintf("Does \$state include BAR state? %s\n", (bool)($state & STATE_BAR));
echo sprintf("Does \$state include FEZ state? %s\n", (bool)($state & STATE_FEZ));
echo sprintf("Does \$state include BAZ state? %s\n", (bool)($state & STATE_BAZ));
echo sprintf("Is state equivalent to FOO and FEZ states? %s\n", ($state == (STATE_FOO | STATE_FEZ)));

Output:

STATE_FOO's base2 value is 00000001
STATE_BAR's base2 value is 00000010
STATE_FEZ's base2 value is 00000100
STATE_BAZ's base2 value is 00001000

base10 value of $state is 5
base2 value of $state is 00000101
Does $state include FOO state? 1
Does $state include BAR state?
Does $state include FEZ state? 1
Does $state include BAZ state?
Is state equivalent to FOO and FEZ states? 1

Forget what is already in your head.

OK, now say you have some different roles: admin, user, and guest.

and some different permissions: read, write and delete

Let's create some bitmasks for permissions and roles. A bitmask is a sequence of bits that can be used to manipulate or read some kind of flags. As shown below:

// flags                                bitmasks
$read = 1;                              // 0001
$write = 2;                             // 0010
$delete = 4;                            // 0100

$admin = $read | $write | $delete;      // 0001 | 0010 | 0100 => 0111
$user = $read | $write;                 // 0001 | 0010 => 0011 
$guest = $read;                         // 0001 => 0001

Notice 1, 2, 4. This must be raised as double. Otherwise, it might give you some awkward results.

Forget about the things commented. Those are just sequence of bits (or bitmasks) for individual permissions and roles.

Now let's create a handy function which may be used to check a specific permission for a specific role.

function isAllowed($role, $permissison) {
    return $role & $permissison ? true : false;
}

We are done. Let's check the $delete permission for all 3 roles:

var_dump(isAllowed($admin, $delete));  // bool(true)
var_dump(isAllowed($user, $delete));   // bool(false)
var_dump(isAllowed($guest, $delete));  // bool(false)

So why is bitwise operation? In a word, bitwise operation is more faster, concise and maintainable. Otherwise, using bitwise operation is always efficient for complex applications.