English is not my 1st language. You have 2 options: "Apple, betrayed by its own law firm" or "Apple was betrayed by its own law firm", you cannot just say "Apple betrayed by its own law firm".
There seem to be different conventions for headlines because they are not always complete sentences. For example, many newspapers would be fine with "Man killed by unknown assailant" as a headline, or "Budget increase passed by legislature", which have the same grammatical structure as "Apple betrayed by its own law firm".
Edit: oops, but that's not what this headline did. Nonetheless, the headline seems reasonable to me as a native speaker.
Not the same structure at all. "Apple betrayed by its own law firm" is passive voice; your example is active voice. Passive voice in English is constructed with "to be" as the conjugated "helping verb" and the past participle of the actual verb. This is also why the law firm must be prefaced with "by"; the actor is not included in passive voice. earlyriser is correct in his proposed insertion of "was".
The equivalent to your example would be, "Law firm betrayed Apple, its client."